<p><Success style='float: center;margin-top:16rem;text-align:center'><B>Quadratic equations lecture - 2</B></Success></p>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Sum and product of quadratic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>suppose $\alpha, \beta \rightarrow$ roots</p> <p>$x=\alpha$</p> <p>$x-\alpha$ will be factor of quadratic equation</p> <p>$a x^2+bx+c=0$</p> <p>$x=\beta$</p> <p>$x-\beta$ will also be factor of $a x^2+b x+c=0$</p> </div> </main> <hr> <footer style='font-size:18px'> <span style="color:blue">Sum and product of quadratic equation </span> $\to$ Example $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Sum and product of quadratic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$(x- \alpha) (x- \beta)=0$</p> <p>$x^2- \beta x - \alpha x +\alpha\beta=0$</p> <p>$x^2- ( \alpha + \beta )x +\alpha\beta=0$</p> <p>$x^2- (sum \ \ of \ \ roots)x +product \ \ of \ \ roots=0$</p> </main> <hr> <footer style='font-size:18px'> <span style="color:blue">Sum and product of quadratic equation </span> $\to$ Example $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$2+ \sqrt3, 2- \sqrt3$</p> <p>$Now, \alpha+ \beta=2 + \sqrt3 +2- \sqrt3 = 4$</p> <p>$Now, \alpha \beta=(2 + \sqrt3) (2- \sqrt3)$</p> <p>$=2^2-(\sqrt3)^2$</p> <p>$=4-3 = 1$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue"> Example</span> $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Quadratic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$x^2- ( \alpha + \beta )x +\alpha\beta=0$</p> <p>$x^2- 4x+1=0$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ <span style="color:blue">Quadratic equation </span>$\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$\alpha=2+i$</p> <p>$\beta=2-i$</p> <p>$\alpha+\beta=2+i+2-i = 4$</p> <p>$\alpha\beta=(2+i).(2-i)$</p> <p>$=2^2-(i)^2 = 4+1 = 5$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue">Example </span>$\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Quadratic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$x^2- ( \alpha + \beta )x +\alpha\beta=0$</p> <p>$x^2- 4x+5=0$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$<span style="color:blue"> Quadratic equation</span> $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Transformation of quadratic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ax^2+bx+c=0$</p> <p>$put \ \ x= \frac{1}{y}$</p> <p>Equation,</p> <p>$a( \frac{1}{y})^2+b( \frac{1}{y})+c=0$</p> <p>$ \Rightarrow \frac{a}{y^2}+ \frac{b}{y}+c=0$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ Quadratic equation $\to$<span style="color:blue"> Transformation of quadratic equation </span>$\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Transformation of quadratic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ \Rightarrow y^2[ \frac{a}{y^2}+ \frac{b}{y}+c]=0 \times y^2$</p> <p>$ \Rightarrow a+by+cy^2=0$</p> <p>$ \Rightarrow cy^2+by+a=0$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ Quadratic equation $\to$ <span style="color:blue">Transformation of quadratic equation</span> $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ x^2+7x+12=0$</p> <p>$put \ \ x=1/y$</p> <p>$ \Rightarrow \frac{1}{y^2}+7( \frac{1}{y})+12=0$</p> <p>$\Rightarrow \frac{1}{y^2}+\frac{7}{y}+12=0$</p> <p>$\Rightarrow 12y^2+7y+1=0$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue">Example </span>$\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ x^2+7x+12=0$</p> <p>$ \Rightarrow x^2+(4+3)x+12=0$</p> <p>$ \Rightarrow x^2+4x+3x+12=0$</p> <p>$\Rightarrow x(x+4)+3(x+4)=0$</p> <p>$\Rightarrow (x+4) \ (x+3)=0$</p> <p>$\Rightarrow x=-3,-4$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue">Example </span>$\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ 12y^2+7y+1=0$</p> <p>Now we shall find out the root of this quadratic equation should be 1/-3,1/-4.</p> <p>Let us try to find out the root of the</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue"> Example </span>$\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ 12y^2+7y+1=0$</p> <p>$ \Rightarrow 12y^2+(4+3)y+1=0$</p> <p>$ \Rightarrow 12y^2+4y+3y+1=0$</p> <p>$\Rightarrow 4y(3y+1)+1(3y+1)=0$</p> <p>$\Rightarrow (3y+1) \ (4y+1)=0$</p> <p>$\Rightarrow y=\frac{-1}{3},\frac{-1}{4}$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue">Example </span>$\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Transformation of quadratic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ ax^2+bx+c=0$</p> <p>$ put \ x=-y$</p> <p>$ \Rightarrow a(-y)^2+b(-y)+c=0$</p> <p>$ \Rightarrow ay^2-by+c=0$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ Quadratic equation $\to$<span style="color:blue"> Transformation of quadratic equation</span> $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ x^2-3x+2=0$</p> <p>$ \Rightarrow \ x=-y$</p> <p>transformed quadratic equation</p> <p>$ \Rightarrow (-y)^2-3(-y)+2=0$</p> <p>$ \Rightarrow y^2+3y+2=0$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue"> Example</span> $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ x^2-3x+2=0$</p> <p>$ \Rightarrow x^2-(2+1)x+2=0$</p> <p>$ \Rightarrow x^2-2x-x+2=0$</p> <p>$ \Rightarrow x(x-2)-1(x-2)=0$</p> <p>$ \Rightarrow (x-2) \ (x-1)$</p> <p>$ \Rightarrow x=1,2 $</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue"> Example </span>$\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ y^2+3y+2=0$</p> <p>$ \Rightarrow y^2+(2+1)y+2=0$</p> <p>$ \Rightarrow y^2+2y+y+2=0$</p> <p>$ \Rightarrow y(y+2)+1(y+2)=0$</p> <p>$ \Rightarrow (y+2) \ (y+1)$</p> <p>$ \Rightarrow y=-2,-1 $</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue">Example</span> $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ x^2-3x+2=0 \rightarrow $ (1,2)</p> <p>$ \Rightarrow \ x=-y$</p> <p>transformed quadratic equation</p> <p>$ \Rightarrow (-y)^2-3(-y)+2=0$</p> <p>$ \Rightarrow y^2+3y+2=0 \rightarrow $ (-1,-2)</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue">Example</span> $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Transformation of quadratic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ \Rightarrow ax^2+bx+c=0$</p> <p>$ \Rightarrow put \ \ x= \sqrt y$</p> <p>$ \Rightarrow a(\sqrt y)^2+b(\sqrt y)+c=0$</p> <p>$ \Rightarrow ay+b\sqrt y+c=0$</p> <p>$ \Rightarrow ay+c=-b \sqrt y$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ Quadratic equation $\to$ <span style="color:blue">Transformation of quadratic equation </span>$\to$ Solution $\to$ Cubic equation </footer> <p>======</p> <p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ \Rightarrow \text{squaring both sides}$</p> <p>$ \Rightarrow (ay+c)^2=b^2y$</p> <p>$ \Rightarrow a^2y^2+c^2+2acy=b^2y$</p> <p>$ \Rightarrow a^2y^2+(2ac-b^2)y+c^2=0$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue">Example </span>$\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer> <p>======</p> <p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ x^2+7x+12=0$</p> <p>$ put \ x= \sqrt y$</p> <p>$ \Rightarrow a(\sqrt y)^2+7(\sqrt y)+12=0$</p> <p>$ \Rightarrow y+7 \sqrt y+12=0$</p> <p>$ \Rightarrow (y+12)=-7 \sqrt y$</p> <p>$ \Rightarrow \text{squaring both sides}$</p> <p>$ y^2+144+24y=49y$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue"> Example</span> $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Quadratic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ y^2-25y+144=0$</p> <p>Now find the root of the first quadratic equation</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ <span style="color:blue">Quadratic equation</span> $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Quadratic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ \Rightarrow x^2+7x+12=0$</p> <p>$ \Rightarrow x^2+(4+3)x+12=0$</p> <p>$ \Rightarrow (x+4) \ (x+3)=0$</p> <p>$ \Rightarrow x=-3,-4$</p> <p>$ \Rightarrow y^2-25y+144=0$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$<span style="color:blue"> Quadratic equation</span> $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Quadratic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ \Rightarrow y^2-(16+9)y+144=0$</p> <p>$ \Rightarrow y^2-16y-9y+144=0$</p> <p>$ \Rightarrow y(y-16)-9(y-16)=0$</p> <p>$ \Rightarrow (y-16) (y-9)=0$</p> <p>$ \Rightarrow y=16,9$</p> <p>$ \Rightarrow x=-3,-4$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ <span style="color:blue">Quadratic equation</span> $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Quadratic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:60%;font-size:22px;'> <p>$ (I) \to a_1x^2+b_1x+c_1=0$</p> <p>$ (II) \to a_2x^2+b_2x+c_2=0$</p> <p>$\alpha \text{ is the common root}$</p> <p>$ a_1\alpha^2+b_1\alpha+c_1=0$</p> <p>$ \Rightarrow a_2\alpha^2+b_2\alpha+c_2=0$</p> </div> <div style='float: left;text-align:left; width:40%;font-size:22px;'> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><msup><mi>α</mi><mn>2</mn></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>b</mi><mn>1</mn></msub></mtd><mtd><msub><mi>c</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mn>2</mn></msub></mtd><mtd><msub><mi>c</mi><mn>2</mn></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">|</mo></mrow></mfrac><mo>=</mo><mfrac><mi>α</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>c</mi><mn>1</mn></msub></mtd><mtd><msub><mi>a</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>c</mi><mn>2</mn></msub></mtd><mtd><msub><mi>a</mi><mn>2</mn></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">|</mo></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>a</mi><mn>1</mn></msub></mtd><mtd><msub><mi>b</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>a</mi><mn>2</mn></msub></mtd><mtd><msub><mi>b</mi><mn>2</mn></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">|</mo></mrow></mfrac></math></p> <p>$\frac{\alpha^2}{\left(b_1 c_2-b_2 c_1\right)}=\frac{\alpha}{c_1 a_2-c_2 a_1}=\frac{1}{a_1 b_2-a_2 b_1}$</p> <p>$ \alpha^2=\frac{b_1 c_2-b_2 c_1}{a_1 b_2-a_2 b_1} $</p> <p>$ \alpha=\frac{c_1 a_2-c_2 a_1}{a_1 b_2-a_2 b_1}$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ <span style="color:blue">Quadratic equation</span> $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Quadratic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ \alpha^2=(\alpha)^2$</p> <p>$ \frac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1}= (\frac{c_1a_2-c_2a_1}{a_1b_2-a_2b_1})^2$</p> <p>$\Rightarrow \frac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1}=\frac{(c_1a_2-c_2a_1)^2}{(a_1b_2-a_2b_1)^2}$</p> <p>$\Rightarrow (c_1a_2-c_2a_1)^2= (b_1c_2-b_2c_1) \times (a_1b_2-a_2b_1)$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ <span style="color:blue">Quadratic equation</span> $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>Find the value of k, if the equations 2x²+kx-5=0 and x²-3x-4=0 have one roots in common.</p> </div> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue"> Example</span> $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer> <p>======</p> <p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>α be the common root.</p> <p>$ \Rightarrow 2\alpha^2+k\alpha-5=0 $</p> <p>$ \Rightarrow \alpha^2-3\alpha-4=0 $</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ <span style="color:blue">Solution</span> $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ \frac{\alpha^2}{-4k-15}=\frac{\alpha}{-5+8}=\frac{1}{-6-k}$</p> <p>$ \frac{\alpha^2}{-4k-15}=\frac{\alpha}{3}=\frac{1}{-6-k}$</p> <p>$ \alpha^2=\frac{4k+15}{k+6},\alpha=\frac{-3}{k+6}$</p> <p>$ \Rightarrow \alpha^2=\alpha^2$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$<span style="color:blue"> Solution</span> $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ \frac{4k+15}{k+6}=\frac{9}{(k+6)^2}$</p> <p>$ \Rightarrow 4k^2+24k+15k+90=9$</p> <p>$ \Rightarrow 4k^2+39k+81=0$</p> <p>$ \Rightarrow 4k^2+27k+12k+81=0$</p> <p>$ \Rightarrow k(4k+27)+3(4k+27)=0$</p> <p>$ \Rightarrow k= \frac{-27}{4},-3$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ <span style="color:blue">Solution</span> $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>If a,b,c are +ve real numbers such that the equations $ax^2+bx+c=0$ and $bx^2+cx+a=0$ have common root, then find the relation in among a,b and c.</p> </div> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue">Example</span> $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer> <p>======</p> <p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$\alpha$ be the common root</p> <p>$ \Rightarrow a \alpha^2+b \alpha+c=0 \longrightarrow (1) $</p> <p>$ \Rightarrow b \alpha^2+c \alpha+a=0 \longrightarrow(2)$</p> <p>$ \frac{\alpha^2}{a b-c^2}=\frac{\alpha}{c b-a^2}=\frac{1}{a c-b^2} $</p> <p>$\alpha^2=\frac{a b-c^2}{a c-b^2} $</p> <p>$\alpha=\frac{c b-a^2}{a c-b^2} $</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ <span style="color:blue">Solution</span> $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:90%;font-size:22px;'> <p>$ \Rightarrow \alpha^2=(\alpha)^2$</p> <p>$ \Rightarrow \frac{ab-c^2}{ac-b^2}=(\frac{cb-a^2}{ac-b^2})^2$</p> <p>$ \Rightarrow (ab-c^2)(ac-b^2)=(cb-a^2)^2$</p> <p>$ \Rightarrow a^2bc-ab^3-ac^3+b^2c^2 = b^2c^2+a^4-2a^2bc$</p> <p>$\Rightarrow 3 a^2 b c=a^4+a b^3+a c^3$</p> <p>$ \Rightarrow a^3+b^3+c^3=3abc$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$<span style="color:blue"> Solution</span> $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Cubic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:100%;font-size:22px;'> <p>$ a_0x^3+a_1x^2+a_2^x+a_3=0$</p> <p>$ a_0 \neq0 \ \ ,a_0,a_1,a_2,a_3 \epsilon R$</p> <p>$\rArr$ Relation between roots of cubic equation and its coefficient.</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ <span style="color:blue">Cubic equation</span> </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Cubic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:100%;font-size:22px;'> <p>$ \Rightarrow a_0x^3+a_1x^2+a_2x+a_3=0 $</p> <p>$ \alpha,\beta,\gamma \to roots $</p> <p>$ \Rightarrow a_0x^3+a_1x^2+a_2x+a_3=a_0(x-\alpha)(x-\beta)(x-\gamma)$</p> <p>$ \Rightarrow a_0x^3+a_1x^2+a_2x+a_3=a_0[(x^2-(\alpha+\beta)x+\alpha\beta] (x-\gamma)$</p> </div> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ <span style="color:blue">Cubic equation</span> </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Cubic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:100%;font-size:22px;'> <p>$ \Rightarrow a_0x^3+a_1x^2+a_2x+a_3=a_0[(x^3-\gamma x^2-(\alpha+\beta)x^2+(\alpha+\beta).\gamma x+\alpha\beta x-\alpha\beta\gamma]$</p> <p>$=a_0[(x^3-(\alpha+\beta+\gamma)x^2+(\alpha\beta+\beta\gamma+\gamma\alpha)x-\alpha\beta\gamma]$</p> </div> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ <span style="color:blue">Cubic equation</span> </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Cubic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:100%;font-size:22px;'> <p>$ \Rightarrow a_0x^3+a_1x^2+a_2x+a_3=a_0x^3-a_0(\alpha+\beta+\gamma)x^2+a_0(\alpha\beta+\beta\gamma+\gamma\alpha)x-a_0 \alpha\beta\gamma$</p> <p>$ \Rightarrow x^2 \to $</p> <p>$ a_1=-a_0(\alpha+\beta+\gamma)$</p> <p>$ S_1=\alpha+\beta+\gamma=-\frac{a_1}{a_0}$</p> </div> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ <span style="color:blue">Cubic equation</span> </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Cubic equation</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:100%;font-size:22px;'> <p>$x \to a_2=a_0(\alpha\beta+\beta\gamma+\gamma\alpha)$</p> <p>$ S_2=\alpha\beta+\beta\gamma+\gamma\alpha= \frac{a_2}{a_2}$</p> <p>$ conc.\ S_3=\alpha\beta\gamma=-\frac{a_3}{a_0}$</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ Example $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ <span style="color:blue">Cubic equation</span> </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:100%;font-size:22px;'> <p>Find the value of $\alpha+\beta+\gamma$, $\alpha\beta+\beta\gamma+\gamma\alpha$ and $\alpha\beta\gamma$ if $\alpha,\beta,\gamma$ are the roots of cubic equation $x^3+6x^2+5x-12=0$</p> <p>$ sol: a_0x^3+a_1x^2+a_2x+a_3=0$</p> <p>$ a_0=1, a_1=6,a_2=5,a_3=-12$</p> <p>$ \alpha+\beta+\gamma=-\frac{a_1}{a_0}$</p> <p>$ =-\frac{6}{1} $</p> <p>$ =-6 $</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue"> Example </span>$\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equations lecture - 2</B></Success></p> <p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example</B></Primary></p> <hr> <main> <div style='float: left;text-align:left; width:100%;font-size:22px;'> <p>$ \alpha\beta+\beta\gamma+\gamma\alpha=-\frac{a_2}{a_0}$</p> <p>$ =\frac{5}{1} $</p> <p>$ =5 $</p> <p>$ \alpha\beta+\beta\gamma+\gamma\alpha=-\frac{a_3}{a_0}$</p> <p>$ =+\frac{12}{1} $</p> <p>$ =12 $</p> </div> </main> <hr> <footer style='font-size:18px'> Sum and product of quadratic equation $\to$ <span style="color:blue">Example</span> $\to$ Quadratic equation $\to$ Transformation of quadratic equation $\to$ Solution $\to$ Cubic equation </footer>

<p><Success style='float: center;margin-top:16rem;text-align:center'><B>Thank you</B></Success></p>