Question

Find the condition that
one of the roots of the equation ax2 + bx + c = 0
(i)may be unity,
(ii) one of the roots of the equation ax2 + bx + c = 0 may be zero
(ii) exactly one of the roots of the equation ax2 + bx + c = 0 may be zero
​

Answer 1

(i)

The given quadratic equation is

$\quad ax^{2}+bx+c=0$

Since one of the roots of this equation is $1$, we put $x=1$ in the above equation.

$\Rightarrow a(1)^{2}+b(1)+c=0$

$\Rightarrow a+b+c=0$

This is the required condition.

(ii)

The given quadratic equation is

$\quad ax^{2}+bx+c=0$

Since one of the roots of this equation is $0$, we put $x=0$ in the above equation.

$\Rightarrow a(0)^{2}+b(0)+c=0$

$\Rightarrow c=0$

This is the required condition.

(iii)

The given quadratic equation is

$\quad ax^{2}+bx+c=0$

Since exactly of the roots of this equation is $0$, we find the solutions of the above equation first, which are

$\quad x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$

Since exactly one root may be $0$, then

either $\frac{-b+\sqrt{b^{2}-4ac}}{2a}=0$

$\quad\Rightarrow -b+\sqrt{b^{2}-4ac}=0$

$\quad\Rightarrow -b=-\sqrt{b^{2}-4ac}$

$\quad\Rightarrow b^{2}=b^{2}-4ac$

$\quad\Rightarrow ac=0$

or $\frac{-b-\sqrt{b^{2}-4ac}}{2a}=0$

$\quad\Rightarrow -b-\sqrt{b^{2}-4ac}=0$

$\quad\Rightarrow -b=\sqrt{b^{2}-4ac}$

$\quad\Rightarrow b^{2}=b^{2}-4ac$

$\quad\Rightarrow ac=0$

Thus the required condition is $ac=0$.

Note:

In (iii), though the obtained condition is $ac=0$, $a\neq 0$, since if $a=0$, the given equation will no longer be quadratic.