If the roots of $a x^{2} - b x - c = 0$ change by the same quantity, then the expression in $a, b, c$ that does not change is

\frac{b^{2} - 4 a c}{a^{2}}

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\frac{b - 4 c}{a}

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\frac{b^{2} + 4 a c}{a^{2}}

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none of these

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Solution

The correct option is D $\frac{b^{2} + 4 a c}{a^{2}}$
Roots of the given equation is $\frac{b \pm \sqrt{b^{2} + 4 a c}}{2 a}$

Let the roots be $α, β$

Let we increase the roots by "k" (some constant)

Therefore, roots become $(α + k, β + k)$

Now $α, β$ satisfy the above quadratic equation ;let $α + k = x$ $\to$ $α = x - k$

Substituting $α$ in above equation ;

$\to a {(x - k)}^{2} - b (x - k) - c = 0$

$\to x^{2} (a) - x (2 a k + b) + a k^{2} + b k - c = 0$

We will check the descriminant of above equation ;

$\to \frac{(2 a k + b) \pm \sqrt{4 a^{2} k^{2} + b^{2} + 4 a b k - 4 a^{2} k^{2} - 4 a b k + 4 a c}}{2 a}$

$\Rightarrow \frac{(2 a k + b) \pm \sqrt{b^{2} + 4 a c}}{2 a}$

We can see descriminant quantity $\frac{\sqrt{b^{2} + 4 a c}}{2 a}$ is same in both equations

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If the roots of ax2−bx−c=0 change by the same quantity, then the expression in a,b,c that does not change is

If the roots of $a x^{2} - b x - c = 0$ change by the same quantity, then the expression in $a, b, c$ that does not change is