First slide

Theory of equations

Question

$The equation formed by decreasing each root of a x^{2} + b x + c = 0 by 1 is 2 x^{2} + 8 x + 2 = 0, then$

Moderate

A

$a = - b$
B

$b = - c$
C

$c = - a$
D

$b = a + c$

Solution

$Let α, β be the roots of the given equation a x^{2} + b x + c = 0$
$∴ α + β = - \frac{b}{a}; α β = \frac{c}{a}$
$New roots of equation 2 x^{2} + 8 x + 2 = 0 are α - 1, β - 1$
$Their sum = α + β - 2 = - \frac{8}{2} - \frac{b}{a} - 2 = - 4 \Rightarrow \frac{b}{a} = 2$
$\begin{array}{l} \begin{aligned} Their product & = (α - 1) (β - 1) = 1 \\ α β - (α + β) + 1 = 1 \\ \frac{c}{a} + \frac{b}{a} = 0 \\ \frac{c}{a} + 2 = 0 \end{aligned} \\ \frac{c}{a} = - 2 = - \frac{b}{a} \end{array}$
$\Rightarrow b = - c$

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