Question

If the difference between the roots of $a{x}^2+bx+c=0$ is 1 , then which one of the following is correct ?

• A. $b^2=a(a+4c)$
• B. $a^2=b(b+4c)$
• C. $a^2=c(a+4c)$
• D. $b^2=a(b+4c)$

Answer 1

The correct option is A $b^2=a(a+4c)$

Let $p,q$ be roots of equation $a{x}^2+bx+c=0$

So, $p+q=\frac{-b}{a}$ and $pq=\frac{c}{a}$

Difference between roots is given $1$

So, $p-q=1$

$\Rightarrow 1=\sqrt{{(p+q)}^2-4pq}\Rightarrow 1=\sqrt{{\left(\frac{-b}{a}\right)}^2-4\left(\frac{c}{a}\right)}\Rightarrow 1=\sqrt{\frac{b^2-4ac}{a^2}}\Rightarrow a^2=b^2-4ac\Rightarrow b^2=a(a+4c)$