mathematics-class-11-unit-05-chapter-01-quadratic-equations-l-1-15-w8i-uovqabs

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equation  lecture-1</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B> Discriminant of quadratic equation</B></Primary></p>
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<p>$ a x^2+b x+c=0 $</p>
<p>$ \Leftrightarrow a\left(x^2+\frac{b}{a} x+\frac{c}{a}\right)=0$</p>
<p>$ \Leftrightarrow a\left(x^2+2 \cdot x \cdot \frac{b}{2 a}+\frac{b^2}{4 a^2}-\frac{b^2}{4 a^2}+\frac{c}{a}\right)$</p>
<p>$ \Leftrightarrow a\left\{\left(x+\frac{b}{2 a}\right)^2-\left(\frac{\sqrt{b^2-4 a c}}{2 a}\right)^2\right\}=0 $</p>
<p>$ D:=b^2-4 a c$</p>
<p>$( \text { Discriminant of the } \text { polynomial } a x^2+b x+c )$</p>
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Quadratic equation $\to$ <span style="color:blue">Discriminant of quadratic equation</span> $\to$ Nature of roots of quadratic equation $\to$ Case I $\to$ Case II $\to$ Case III $\to$ $\text{Sign of}\\ \alpha, \beta$ $\to$ Recall $\to$ Question $\to$ Solution
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equation  lecture-1</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Case I</B></Primary></p>
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<p>$\text { ii) } b<0, c<0$</p>
<p>$\text {iii) } c<0, b>0$</p>
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Quadratic equation $\to$ Discriminant of quadratic equation $\to$ Nature of roots of quadratic equation $\to$ <span style="color:blue">Case I</span> $\to$ Case II $\to$ Case III $\to$ $\text{Sign of}\\ \alpha, \beta$ $\to$ Recall $\to$ Question $\to$ Solution
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equation  lecture-1</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Case II</B></Primary></p>
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<p><strong>Case II: $D = 0$</strong></p>
<p>i) $c>0 , b>0$</p>
<p>ii) $c>0, b>0$</p>
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Quadratic equation $\to$ Discriminant of quadratic equation $\to$ Nature of roots of quadratic equation $\to$ Case I $\to$ <span style="color:blue">Case II </span>$\to$ Case III $\to$ $\text{Sign of}\\ \alpha, \beta$ $\to$ Recall $\to$ Question $\to$ Solution
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<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Case III</B></Primary></p>
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<p><strong>Case III: $D<0 $</strong></p>
<p>$ \text { i) } b<0, c>0$</p>
<p>No real Solution $\alpha, \beta$ are not real numbers</p>
<p>$ \text{ii)}$  $b>0, c>0$</p>
<p>No real solution</p>
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Quadratic equation $\to$ Discriminant of quadratic equation $\to$ Nature of roots of quadratic equation $\to$ Case I $\to$ Case II $\to$ <span style="color:blue">Case III</span> $\to$ $\text{Sign of}\\ \alpha, \beta$ $\to$ Recall $\to$ Question $\to$ Solution
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equation  lecture-1</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Case III</B></Primary></p>
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<p>iii) $b=0, c>0$</p>
<p>No solution</p>
<p>$D=b^2-4 a c<0$</p>
<p>$b^2<4 a c$</p>
<p>$\Rightarrow c>\frac{b^2}{4 a}$</p>
<p>$b^2 \geqslant 0,a>0$</p>
<p>$\Rightarrow \frac{b^2}{4 a} \geqslant 0, c \geqslant 0$</p>
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Quadratic equation $\to$ Discriminant of quadratic equation $\to$ Nature of roots of quadratic equation $\to$ Case I $\to$ Case II $\to$ <span style="color:blue">Case III</span> $\to$ $\text{Sign of}\\ \alpha, \beta$ $\to$ Recall $\to$ Question $\to$ Solution
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equation  lecture-1</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>$\text{Sign of}\ \alpha, \beta$</B></Primary></p>
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<p>if $c>0$, then</p>
<p>$D  =b^2-4 a c<b^2 $</p>
<p>$ \therefore \quad \sqrt{D}<b \quad \text { as } D, b>c$</p>
<p>$\therefore \quad-b+\sqrt{D}<0$</p>
<p>$\therefore \quad  \beta=\frac{-b+\sqrt{D}}{2 a}<0 $</p>
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Quadratic equation $\to$ Discriminant of quadratic equation $\to$ Nature of roots of quadratic equation $\to$ Case I $\to$ Case II $\to$ Case III $\to$ <span style="color:blue">$\text{Sign of}\\ \alpha, \beta$</span> $\to$ Recall $\to$ Question $\to$ Solution
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equation  lecture-1</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>$\text{Sign of}\ \alpha, \beta$</B></Primary></p>
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<p>ii) if $\quad b<0$</p>
<p>$
\beta=\frac{-b+\sqrt{D}}{2 a}>0
$</p>
<p>if $c>0$</p>
<p>$
D=b^2-4 a c<b^2
$</p>
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Quadratic equation $\to$ Discriminant of quadratic equation $\to$ Nature of roots of quadratic equation $\to$ Case I $\to$ Case II $\to$ Case III $\to$<span style="color:blue"> $\text{Sign of}\\ \alpha, \beta$</span> $\to$ Recall $\to$ Question $\to$ Solution
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<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equation  lecture-1</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>$\text{Sign of}\ \alpha, \beta$</B></Primary></p>
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<p>$
\therefore \sqrt{D}<-b ~
$
as $ ~ D>0$</p>
<p>and $ ~ b<0$</p>
<p>$-b-\sqrt{D}>0$</p>
<p>$
\therefore \quad \alpha=\frac{-b-\sqrt{D}}{2 a}>0
$</p>
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Quadratic equation $\to$ Discriminant of quadratic equation $\to$ Nature of roots of quadratic equation $\to$ Case I $\to$ Case II $\to$ Case III $\to$ <span style="color:blue">$\text{Sign of}\\ \alpha, \beta$</span> $\to$ Recall $\to$ Question $\to$ Solution
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<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p>
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<p>$y=2+\sqrt{5}$</p>
<p>$e^{\sin x}=2+\sqrt{5}$</p>
<p>$\therefore ~ \sin x=\log (2+\sqrt{5})$</p>
<p>$2+\sqrt{5}>4>e$</p>
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Quadratic equation $\to$ Discriminant of quadratic equation $\to$ Nature of roots of quadratic equation $\to$ Case I $\to$ Case II $\to$ Case III $\to$ $\text{Sign of}\\ \alpha, \beta$ $\to$ Recall $\to$ Question $\to$ <span style="color:blue">Solution</span>
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<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Quadratic equation  lecture-1</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p>
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<p>$\therefore \log (2+\sqrt{5})>\log e=1$</p>
<p>$\therefore \quad \sin x>1$</p>
<p>$ x \in \mathbb{R} $</p>
<p>$ -1 \leqslant \sin x \leqslant 1$</p>
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Quadratic equation $\to$ Discriminant of quadratic equation $\to$ Nature of roots of quadratic equation $\to$ Case I $\to$ Case II $\to$ Case III $\to$ $\text{Sign of}\\ \alpha, \beta$ $\to$ Recall $\to$ Question $\to$ <span style="color:blue">Solution</span>
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<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p>
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<p>$\therefore \frac{a^2+b^2+c^2}{a b+b c+c a}<2$</p>
<p>$\frac{a^2+b^2+c^2}{a b+b c+c a}\geqslant 3 \lambda-2$</p>
<p>$3 \lambda-2  <2 $</p>
<p>$\therefore \quad \lambda <\frac{4}{3}$</p>
<p>$\lambda=0$</p>
<p>$x^2+2(a+b+c) x=0$</p>
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Quadratic equation $\to$ Discriminant of quadratic equation $\to$ Nature of roots of quadratic equation $\to$ Case I $\to$ Case II $\to$ Case III $\to$ $\text{Sign of}\\ \alpha, \beta$ $\to$ Recall $\to$ Question $\to$ <span style="color:blue">Solution</span>
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