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:

Easy

A

$a = - b$
B

$b = - c$
C

$c = - a$
D

$b = a + c$

Solution

Let roots of $a x^{2} + b x + c = 0$ are $α$ and $β$ then
                  $α + β = - \frac{b}{a}$ and $α β = \frac{c}{a}$                                            ……(i)

then roots of $2 x^{2} + 8 x + 2 = 0$ are

           $(α - 1)$ and $(β - 1)$ then

          $α - 1 + β - 1 = - \frac{8}{2} = - 4$

$\Rightarrow$          $α + β = - 2 \Rightarrow - \frac{b}{a} = - 2$      (From (i))

$∴$               $b = 2 a$

and         $(α - 1) (β - 1) = \frac{2}{2}$

$\Rightarrow$         $α β - α - β + 1 = 1$

$\Rightarrow$             $α β = α + β$

$\Rightarrow$               $\frac{c}{a} = - \frac{b}{a}$              (From (i))

$∴$          $b = - c$

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