Solving a quadratic inequality

3-step method

1. Rewrite the inequality in the form ax^2 + bx + c >= 0 (or <=, >, <), i.e. with all terms on one side and 0 on the other.
2. Calculate Delta and any roots.
3. Apply the sign rule and consult the sign table to answer the question.

Complete example

Solve: x² - x - 6 ≤ 0

Step 1: Already in the correct form, a = 1, b = -1, c = -6.

Step 2: Delta = 1 – 4 × 1 × (-6) = 1 + 24 = 25 = 5². Roots: x₁ = (1 – 5)/2 = –2, x₂ = (1 + 5)/2 = 3.

Step 3: a = 1 > 0, so f(x) ≤ 0 between the roots (including the endpoints as this is an open inequality).

Solution: S = [-2; 3]

Another example (negative a)

Solve: -2*x^2 + 8 > 0

This is equivalent to solving -2*x^2 + 8 = 0 -> x^2 = 4 -> x1 = -2, x2 = 2. Here, a = -2 < 0.

f(x) has the same sign as -a (positive) between the roots.

Strict solution: S = ]-2; 2[

Common pitfalls

• Do not forget to group everything on one side of the inequality before calculating Δ.
• Make a clear distinction between open inequalities (<=, >=), which include the roots, and strict inequalities (<, >), which exclude them.
• If Δ < 0 and a > 0, then f(x) ≥ 0 holds for all real x (S = R), whilst f(x) ≤ 0 has no solutions.