12.5 The quadratic formula — Learn

Prerequisites: rearranging equations, square roots, and substituting values into a formula.

Objectives: solve a quadratic equation using the quadratic formula and interpret its discriminant. This covers A18 at Higher level.

The formula

For $ax^2+bx+c=0$, where $a\ne 0$,

The expression $b^2-4ac$ is the discriminant. A positive discriminant gives two real roots, zero gives one repeated root, and a negative discriminant gives no real roots.

Example 1. Solve $2x^2+3x-2=0$.

Here $a=2$, $b=3$, and $c=-2$.

Watch out. Keep the negative sign in $c=-2$ when substituting. The discriminant is $9-4(2)(-2)=25$, not $9-16$.

You should now be able to

• identify $a$, $b$, and $c$ after writing the equation in standard form;
• substitute carefully into the quadratic formula;
• use the discriminant to describe the number of real roots.