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Parin Sir > Quantitative Aptitude (Maths) > Quadratic Equation > Examples 7 to 9 > Examples 7 to 9

Examples 7 to 9

July 22, 2020
Category: Examples 7 to 9

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(7) Find the values of m for which 16x² + mx + 9 = 0 has both equal roots.

Soln. For equal roots, we must have

D = 0.

$\therefore$ b² – 4 a c = 0

$\therefore$ m² – 576 = 0

$\therefore$ m = $\pm$ 24

(8) If , are the roots of 5x² + 9x + 1 = 0, find the values of :

(i) $\alpha$ ² + $\beta$ ² (ii) $\frac{1}{\alpha}+\frac{1}{\beta}$ (iii) $\alpha$ ³ + $\beta$ ³(iv) $\alpha$ – $\beta$

Soln. We have : $\alpha$ + $\beta$ = $\frac{-9}{5}$ and $\alpha$ $\beta$ = $\frac{1}{5}$ .

(i) $\alpha$ ² + $\beta$ ² = ( $\alpha$ + $\beta$ )² – 2 $\alpha$ $\beta$ = $\frac{79}{25}$

(ii) $\frac{1}{\alpha}+\frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta}$ = – 9

(iii) $\alpha$ ³ + $\beta$ ³ = ( $\alpha$ + $\beta$ )³ – 3 $\alpha$ $\beta$ ( $\alpha$ + $\beta$ ) = – $\frac{594}{125}$

(iv) $\pm \sqrt{(\alpha+\beta)^{2}-4\alpha\beta}$ = $\pm \sqrt{\frac{77}{25}}$

(9) If , are the roots of 4x² + 5x – 1 = 0, form an equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ .

Soln. We have : $\alpha + \beta =$ $\frac{-5}{4}$ and $\alpha$ $\beta$ = $\frac{-1}{4}$ .

Here, $\alpha$ ₁ = $\frac{\alpha}{\beta}$ & $\beta$ ₁ = $\frac{\beta}{\alpha}$

$\alpha$ ₁ + $\beta$ ₁ = $\frac{\alpha}{\beta}$ + $\frac{\beta}{\alpha}$

= $\frac{\alpha ^{2} + \beta^{2}}{\alpha\beta}$

= $\frac{(\alpha+ \beta)^{2}- 2 \alpha \beta}{\alpha \beta}$

= – $\frac{33}{4}$

$\alpha$ ₁· $\beta$ 1

= $\frac{\alpha}{\beta}$ $\times$ $\frac{\beta}{\alpha}$

= 1.

$\therefore$ Required quadratic equation is : x² – ( $\alpha$ ₁ + $\beta$ ₁) x + $\alpha$ ₁· $\beta$ 1 = 0

$\therefore$ x² – $\left (-\frac{33}{4} \right )x$ + 1 = 0

$\therefore$ 4x² + 33 x + 4 = 0

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