First slide

Theory of equations

Question

In writing an equation of the form ${ax}^{2} + bx + c = 0$ ;the coefficient of x is written incorrectly and roots are found to be equal. Again, io writing the same equation the constant term is written incorrectly and it is found that one root is equal to those of the previous wrong equation while the other is double of it. If $α$ and $β$ be the roots of correct equation, then $(α - β)^{2}$ is equal to

Moderate

A

5
B

5 $α β$
C

-4 $α β$
D

-4

Solution

Let the correct equation is ${ax}^{2} + bx + c = 0$
then $α + β = - \frac{b}{a} and αβ = \frac{c}{a}$
When b is written incorrectly, then the roots are equal.
Let these are y and Y
$∴ γ \cdot γ = \frac{c}{a} \Rightarrow γ^{2} = αβ . . . . . . . . (i)$
When c is written incorrectly, then the roots are y and 2y.
$∴ γ + 2 γ = - \frac{b}{a} \Rightarrow 3 γ = α + β$
$\begin{array}{r} \Rightarrow 9 γ^{2} = (α + β)^{2} \Rightarrow 9 αβ = (α - β)^{2} + 4 αβ \\ [from Eq. (i)] \end{array}$
$∴ (α, - β)^{2} = 5 αβ$

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