Review question

# When does this cubic equation have three 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: R6703

## Suggestion

Prove, by considering the turning points on the graph $y=x^3-3a^2x+b$ or otherwise, that if the equation $x^3-3a^2x+b=0$ has three distinct real roots, then $4a^6>b^2$.

Where are the turning points of $y=x^3-3a^2x+b$, and how many are there?

What is the effect of $a$ being negative rather than positive? What happens when $a=0$?

What happens to the number of roots as we translate the curve in the $y$-direction?