The Quadratic Formula

The Quadratic Formula

The quadratic formula is a method used to solve any quadratic equation. The quadratic formula is derived from the process of completing the square, and its use is very flexible in that it can provide solutions for most quadratic equations.

Key points:

• The quadratic formula is given by: x = [ -b ± sqrt(b² - 4ac) ] / 2a
• The formula applies to any equation of the form ax² + bx + c = 0, where a ≠ 0.
• The expression b² - 4ac inside the square root part of the formula is known as the discriminant.
• The value of the discriminant (D) can determine the nature of roots of the quadratic equation.
  • If D > 0, the equation has two distinct real roots.
  • If D = 0, the equation has precisely one real root (or a pair of identical real roots).
  • If D < 0, the equation has no real roots, but two complex roots.

Working with the quadratic formula:

• Check first if you can factor. Not every quadratic equation requires the quadratic formula. Sometimes simple factoring can solve the equation.
• Remember the “±” sign. This indicates that the quadratic formula will always give two solutions.
• Calculate the discriminant first, as this might save time. If discriminant is negative, there’s no need to waste time on the rest of the equation as there will not be a real solution.

Practice Questions

Practise the quadratic formula on a range of quadratic equations to become confident in its application. Use several practise examples, including cases where a is negative and where c is zero. Be sure to meticulously check your work for errors, as one small mistake can throw off the whole solution.