How to find a when the quadratic equation has equal roots?

Question

Let $a$ be a constant. If quadratic equation $(ax-1)^2+a^2 -a-2 = 0$ has equal roots, then $a=$?

Answer 1

Hint:

• A quadratic equation has equal roots iff its discriminant is zero.

• A quadratic equation has equal roots iff these roots are both equal to the root of the derivative.

Answer 2

Hint:

$$(ax-1)^2 = -a^2+a+2$$

It has equal root when $-a^2+a+2=0$.

$$(-a+2)(a+1)=0$$

Answer 3

Your equation can be written as $$a^2x^2-2ax+a^2−a−1=0$$ Just equalize the Discriminant with $0$ i.e. in equation $ax^2+bx+c$ the roots will be equal if $$D=b^2-4ac=0$$.