Review question

# When does $x^2 + ax + a = 1$ have distinct real roots? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9659

## Solution

For what values of the real number $a$ does the quadratic equation $x^2 + ax + a = 1$ have distinct real roots?

1. $a \neq 2$;

2. $a > 2$;

3. $a=2$;

4. all values of $a$.

We can find out about the roots of this quadratic by considering its discriminant.

Before we do this, we must rearrange the equation into the form “$ax^2+bx+c=0$”, so $x^2 + ax + (a-1) = 0.$ Thus the discriminant is equal to $a^2 - 4\times 1 \times (a-1) = a^2 - 4a + 4 = (a-2)^2.$

For the quadratic to have two distinct real roots we need this to be positive and so we need $a \ne 2$.