mathematics-class-11-unit-05-chapter-04-quadratic-equations-l-4-15-5aqklurnjsq

Quadratic equations lecture - 2

Sum and product of quadratic equation

suppose $\alpha, \beta \rightarrow$ roots

$x=\alpha$

$x-\alpha$ will be factor of quadratic equation

$a x^2+bx+c=0$

$x=\beta$

$x-\beta$ will also be factor of $a x^2+b x+c=0$

Quadratic equations lecture - 2

Sum and product of quadratic equation

$(x- \alpha) (x- \beta)=0$

$x^2- \beta x - \alpha x +\alpha\beta=0$

$x^2- ( \alpha + \beta )x +\alpha\beta=0$

$x^2- (sum \ \ of \ \ roots)x +product \ \ of \ \ roots=0$

Quadratic equations lecture - 2

Example

$2+ \sqrt3, 2- \sqrt3$

$Now, \alpha+ \beta=2 + \sqrt3 +2- \sqrt3 = 4$

$Now, \alpha \beta=(2 + \sqrt3) (2- \sqrt3)$

$=2^2-(\sqrt3)^2$

$=4-3 = 1$

Quadratic equations lecture - 2

Quadratic equation

$x^2- ( \alpha + \beta )x +\alpha\beta=0$

$x^2- 4x+1=0$

Quadratic equations lecture - 2

Example

$\alpha=2+i$

$\beta=2-i$

$\alpha+\beta=2+i+2-i = 4$

$\alpha\beta=(2+i).(2-i)$

$=2^2-(i)^2 = 4+1 = 5$

Quadratic equations lecture - 2

Quadratic equation

$x^2- ( \alpha + \beta )x +\alpha\beta=0$

$x^2- 4x+5=0$

Quadratic equations lecture - 2

Transformation of quadratic equation

$ax^2+bx+c=0$

$put \ \ x= \frac{1}{y}$

Equation,

$a( \frac{1}{y})^2+b( \frac{1}{y})+c=0$

$\Rightarrow \frac{a}{y^2}+ \frac{b}{y}+c=0$

Quadratic equations lecture - 2

Transformation of quadratic equation

$\Rightarrow y^2[ \frac{a}{y^2}+ \frac{b}{y}+c]=0 \times y^2$

$\Rightarrow a+by+cy^2=0$

$\Rightarrow cy^2+by+a=0$

Quadratic equations lecture - 2

Example

$x^2+7x+12=0$

$put \ \ x=1/y$

$\Rightarrow \frac{1}{y^2}+7( \frac{1}{y})+12=0$

$\Rightarrow \frac{1}{y^2}+\frac{7}{y}+12=0$

$\Rightarrow 12y^2+7y+1=0$

Quadratic equations lecture - 2

Example

$x^2+7x+12=0$

$\Rightarrow x^2+(4+3)x+12=0$

$\Rightarrow x^2+4x+3x+12=0$

$\Rightarrow x(x+4)+3(x+4)=0$

$\Rightarrow (x+4) \ (x+3)=0$

$\Rightarrow x=-3,-4$

Quadratic equations lecture - 2

Example

$12y^2+7y+1=0$

Now we shall find out the root of this quadratic equation should be 1/-3,1/-4.

Let us try to find out the root of the

Quadratic equations lecture - 2

Example

$12y^2+7y+1=0$

$\Rightarrow 12y^2+(4+3)y+1=0$

$\Rightarrow 12y^2+4y+3y+1=0$

$\Rightarrow 4y(3y+1)+1(3y+1)=0$

$\Rightarrow (3y+1) \ (4y+1)=0$

$\Rightarrow y=\frac{-1}{3},\frac{-1}{4}$

Quadratic equations lecture - 2

Transformation of quadratic equation

$ax^2+bx+c=0$

$put \ x=-y$

$\Rightarrow a(-y)^2+b(-y)+c=0$

$\Rightarrow ay^2-by+c=0$

Quadratic equations lecture - 2

Example

$x^2-3x+2=0$

$\Rightarrow \ x=-y$

transformed quadratic equation

$\Rightarrow (-y)^2-3(-y)+2=0$

$\Rightarrow y^2+3y+2=0$

Quadratic equations lecture - 2

Example

$x^2-3x+2=0$

$\Rightarrow x^2-(2+1)x+2=0$

$\Rightarrow x^2-2x-x+2=0$

$\Rightarrow x(x-2)-1(x-2)=0$

$\Rightarrow (x-2) \ (x-1)$

$\Rightarrow x=1,2$

Quadratic equations lecture - 2

Example

$y^2+3y+2=0$

$\Rightarrow y^2+(2+1)y+2=0$

$\Rightarrow y^2+2y+y+2=0$

$\Rightarrow y(y+2)+1(y+2)=0$

$\Rightarrow (y+2) \ (y+1)$

$\Rightarrow y=-2,-1$

Quadratic equations lecture - 2

Example

$x^2-3x+2=0 \rightarrow$ (1,2)

$\Rightarrow \ x=-y$

transformed quadratic equation

$\Rightarrow (-y)^2-3(-y)+2=0$

$\Rightarrow y^2+3y+2=0 \rightarrow$ (-1,-2)

Quadratic equations lecture - 2

Transformation of quadratic equation

$\Rightarrow ax^2+bx+c=0$

$\Rightarrow put \ \ x= \sqrt y$

$\Rightarrow a(\sqrt y)^2+b(\sqrt y)+c=0$

$\Rightarrow ay+b\sqrt y+c=0$

$\Rightarrow ay+c=-b \sqrt y$

Quadratic equations lecture - 2

Example

$\Rightarrow \text{squaring both sides}$

$\Rightarrow (ay+c)^2=b^2y$

$\Rightarrow a^2y^2+c^2+2acy=b^2y$

$\Rightarrow a^2y^2+(2ac-b^2)y+c^2=0$

Quadratic equations lecture - 2

Example

$x^2+7x+12=0$

$put \ x= \sqrt y$

$\Rightarrow a(\sqrt y)^2+7(\sqrt y)+12=0$

$\Rightarrow y+7 \sqrt y+12=0$

$\Rightarrow (y+12)=-7 \sqrt y$

$\Rightarrow \text{squaring both sides}$

$y^2+144+24y=49y$

Quadratic equations lecture - 2

Quadratic equation

$y^2-25y+144=0$

Now find the root of the first quadratic equation

Quadratic equations lecture - 2

Quadratic equation

$\Rightarrow x^2+7x+12=0$

$\Rightarrow x^2+(4+3)x+12=0$

$\Rightarrow (x+4) \ (x+3)=0$

$\Rightarrow x=-3,-4$

$\Rightarrow y^2-25y+144=0$

Quadratic equations lecture - 2

Quadratic equation

$\Rightarrow y^2-(16+9)y+144=0$

$\Rightarrow y^2-16y-9y+144=0$

$\Rightarrow y(y-16)-9(y-16)=0$

$\Rightarrow (y-16) (y-9)=0$

$\Rightarrow y=16,9$

$\Rightarrow x=-3,-4$

Quadratic equations lecture - 2

Quadratic equation

$(I) \to a_1x^2+b_1x+c_1=0$

$(II) \to a_2x^2+b_2x+c_2=0$

$\alpha \text{ is the common root}$

$a_1\alpha^2+b_1\alpha+c_1=0$

$\Rightarrow a_2\alpha^2+b_2\alpha+c_2=0$

$\frac{α^{2}}{| \begin{matrix} b_{1} & c_{1} \\ b_{2} & c_{2} \end{matrix} |} = \frac{α}{| \begin{matrix} c_{1} & a_{1} \\ c_{2} & a_{2} \end{matrix} |} = \frac{1}{| \begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix} |}$

$\frac{\alpha^2}{\left(b_1 c_2-b_2 c_1\right)}=\frac{\alpha}{c_1 a_2-c_2 a_1}=\frac{1}{a_1 b_2-a_2 b_1}$

$\alpha^2=\frac{b_1 c_2-b_2 c_1}{a_1 b_2-a_2 b_1}$

$\alpha=\frac{c_1 a_2-c_2 a_1}{a_1 b_2-a_2 b_1}$

Quadratic equations lecture - 2

Quadratic equation

$\alpha^2=(\alpha)^2$

$\frac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1}= (\frac{c_1a_2-c_2a_1}{a_1b_2-a_2b_1})^2$

$\Rightarrow \frac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1}=\frac{(c_1a_2-c_2a_1)^2}{(a_1b_2-a_2b_1)^2}$

$\Rightarrow (c_1a_2-c_2a_1)^2= (b_1c_2-b_2c_1) \times (a_1b_2-a_2b_1)$

Quadratic equations lecture - 2

Example

Find the value of k, if the equations 2x²+kx-5=0 and x²-3x-4=0 have one roots in common.

Quadratic equations lecture - 2

Solution

α be the common root.

$\Rightarrow 2\alpha^2+k\alpha-5=0$

$\Rightarrow \alpha^2-3\alpha-4=0$

Quadratic equations lecture - 2

Solution

$\frac{\alpha^2}{-4k-15}=\frac{\alpha}{-5+8}=\frac{1}{-6-k}$

$\frac{\alpha^2}{-4k-15}=\frac{\alpha}{3}=\frac{1}{-6-k}$

$\alpha^2=\frac{4k+15}{k+6},\alpha=\frac{-3}{k+6}$

$\Rightarrow \alpha^2=\alpha^2$

Quadratic equations lecture - 2

Solution

$\frac{4k+15}{k+6}=\frac{9}{(k+6)^2}$

$\Rightarrow 4k^2+24k+15k+90=9$

$\Rightarrow 4k^2+39k+81=0$

$\Rightarrow 4k^2+27k+12k+81=0$

$\Rightarrow k(4k+27)+3(4k+27)=0$

$\Rightarrow k= \frac{-27}{4},-3$

Quadratic equations lecture - 2

Example

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.

Quadratic equations lecture - 2

Solution

$\alpha$ be the common root

$\Rightarrow a \alpha^2+b \alpha+c=0 \longrightarrow (1)$

$\Rightarrow b \alpha^2+c \alpha+a=0 \longrightarrow(2)$

$\frac{\alpha^2}{a b-c^2}=\frac{\alpha}{c b-a^2}=\frac{1}{a c-b^2}$

$\alpha^2=\frac{a b-c^2}{a c-b^2}$

$\alpha=\frac{c b-a^2}{a c-b^2}$

Quadratic equations lecture - 2

Solution

$\Rightarrow \alpha^2=(\alpha)^2$

$\Rightarrow \frac{ab-c^2}{ac-b^2}=(\frac{cb-a^2}{ac-b^2})^2$

$\Rightarrow (ab-c^2)(ac-b^2)=(cb-a^2)^2$

$\Rightarrow a^2bc-ab^3-ac^3+b^2c^2 = b^2c^2+a^4-2a^2bc$

$\Rightarrow 3 a^2 b c=a^4+a b^3+a c^3$

$\Rightarrow a^3+b^3+c^3=3abc$

Quadratic equations lecture - 2

Cubic equation

$a_0x^3+a_1x^2+a_2^x+a_3=0$

$a_0 \neq0 \ \ ,a_0,a_1,a_2,a_3 \epsilon R$

$\rArr$ Relation between roots of cubic equation and its coefficient.

Quadratic equations lecture - 2

Cubic equation

$\Rightarrow a_0x^3+a_1x^2+a_2x+a_3=0$

$\alpha,\beta,\gamma \to roots$

$\Rightarrow a_0x^3+a_1x^2+a_2x+a_3=a_0(x-\alpha)(x-\beta)(x-\gamma)$

$\Rightarrow a_0x^3+a_1x^2+a_2x+a_3=a_0[(x^2-(\alpha+\beta)x+\alpha\beta] (x-\gamma)$

Quadratic equations lecture - 2

Cubic equation

$\Rightarrow a_0x^3+a_1x^2+a_2x+a_3=a_0[(x^3-\gamma x^2-(\alpha+\beta)x^2+(\alpha+\beta).\gamma x+\alpha\beta x-\alpha\beta\gamma]$

$=a_0[(x^3-(\alpha+\beta+\gamma)x^2+(\alpha\beta+\beta\gamma+\gamma\alpha)x-\alpha\beta\gamma]$

Quadratic equations lecture - 2

Cubic equation

$\Rightarrow a_0x^3+a_1x^2+a_2x+a_3=a_0x^3-a_0(\alpha+\beta+\gamma)x^2+a_0(\alpha\beta+\beta\gamma+\gamma\alpha)x-a_0 \alpha\beta\gamma$

$\Rightarrow x^2 \to$

$a_1=-a_0(\alpha+\beta+\gamma)$

$S_1=\alpha+\beta+\gamma=-\frac{a_1}{a_0}$

Quadratic equations lecture - 2

Cubic equation

$x \to a_2=a_0(\alpha\beta+\beta\gamma+\gamma\alpha)$

$S_2=\alpha\beta+\beta\gamma+\gamma\alpha= \frac{a_2}{a_2}$

$conc.\ S_3=\alpha\beta\gamma=-\frac{a_3}{a_0}$

Quadratic equations lecture - 2

Example

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 $x^3+6x^2+5x-12=0$

$sol: a_0x^3+a_1x^2+a_2x+a_3=0$

$a_0=1, a_1=6,a_2=5,a_3=-12$

$\alpha+\beta+\gamma=-\frac{a_1}{a_0}$

$=-\frac{6}{1}$

$=-6$

Quadratic equations lecture - 2

Example

$\alpha\beta+\beta\gamma+\gamma\alpha=-\frac{a_2}{a_0}$

$=\frac{5}{1}$

$=5$

$\alpha\beta+\beta\gamma+\gamma\alpha=-\frac{a_3}{a_0}$

$=+\frac{12}{1}$

$=12$

Thank you