Quadratic Inequalities GCSE: Finding Ranges Where ax² + bx + c > 0

A quadratic inequality looks like ax² + bx + c > 0, just like a quadratic equation but with an inequality sign instead of equals. To solve it, find the roots first, then decide which regions of the curve satisfy the inequality. The answer is usually a range of x values, not a single number.

Why are quadratic inequalities harder than linear ones?

For a linear inequality like 2x + 1 > 5, you rearrange algebraically to get x > 2 — one connected range. With a quadratic, the solution can be two separate regions or one middle range, depending on whether the parabola is above or below zero. You need to use the shape of the parabola, not just algebra, to decide which region.

The golden rule: sketch the parabola (or at least know its shape), find its roots, and then read off the region where the curve satisfies the inequality.

What are the four cases for a quadratic inequality?

Every quadratic inequality with two distinct roots falls into one of these cases, depending on whether the coefficient of x² is positive (U-shaped parabola) or negative (∩-shaped parabola):

where r₁ < r₂ are the two roots. If r₁ = r₂ (repeated root), the inequality ax² + bx + c < 0 has no solutions, and ax² + bx + c > 0 is true for all x ≠ r₁.

How do you solve a quadratic inequality step by step?

Worked example: Solve x² − 5x + 6 > 0.

Step 1: Factorise to find the roots of x² − 5x + 6 = 0.

$$(x - 2)(x - 3) = 0$$

Roots: x = 2 and x = 3. These are the critical values.

Step 2: Sketch the parabola. The coefficient of x² is +1 (positive), so the parabola is U-shaped. It crosses the x-axis at x = 2 and x = 3.

Step 3: Identify the required region. We want x² − 5x + 6 > 0, meaning we want the part of the parabola that is above the x-axis.

For a U-shaped parabola, the curve is above the x-axis outside the roots.

Solution: x < 2 or x > 3

Step 4: Show on a number line with open circles at x = 2 and x = 3 (strict inequality, > not ≥), and arrows pointing left from 2 and right from 3.

How does the solution change for ≥ and ≤?

If the inequality is ≥ or ≤ (rather than strict > or <), the roots are included in the solution:

• x² − 5x + 6 ≥ 0: x ≤ 2 or x ≥ 3 (closed circles on number line).
• x² − 5x + 6 ≤ 0: 2 ≤ x ≤ 3 (closed circles on number line).

The difference: whether the curve is allowed to touch zero at the boundary. With strict inequality, the roots themselves are excluded; with ≥ or ≤, they are included.

Worked example: a < 0 quadratic

Solve −x² + 4x − 3 < 0.

Step 1: Factorise by first factorising x² − 4x + 3 = (x − 1)(x − 3) = 0, then multiply through by −1. The equation −x² + 4x − 3 = 0 has roots x = 1 and x = 3.

Step 2: The coefficient of x² is −1 (negative), so the parabola is ∩-shaped (opens downward). It crosses the x-axis at x = 1 and x = 3.

Step 3: We want −x² + 4x − 3 < 0, meaning below the x-axis. For a ∩-shaped parabola, the curve is below the x-axis outside the roots.

Solution: x < 1 or x > 3

This is the same structure as the U-shape > 0 case — a reminder that you must always check the sign of the x² coefficient and the direction of the inequality together.

How do you write the solution using set notation?

GCSE Higher may ask for set notation:

• "x < 2 or x > 3" in set notation: {x : x < 2} ∪ {x : x > 3}
• "2 < x < 3" in set notation: {x : 2 < x < 3}

The symbol ∪ means "union" (OR — belonging to either set). The colon : means "such that."

At GCSE you are more likely to be asked to write the inequality or show it on a number line, but set notation appears on Higher papers.

Frequently asked questions

What if the quadratic does not factorise neatly?

Use the quadratic formula to find the roots: x = (−b ± √(b² − 4ac)) / (2a). These give the critical values. Then proceed with the sketch and region identification as normal. If the discriminant (b² − 4ac) is negative, the quadratic has no real roots — the parabola never crosses the x-axis, so the inequality is either always true or never true for all real x.

Can I solve a quadratic inequality by dividing by (x − a)?

No — you must not divide or multiply an inequality by an expression containing x, because you do not know whether x − a is positive or negative, and dividing by a negative number reverses the inequality sign. Factorising and sketching is always safe.

How do I show the solution on a number line?

Draw a horizontal number line. Mark the two critical values. Use an open circle (○) for strict inequalities (> or <) and a closed circle (●) for ≥ or ≤. Shade (or draw an arrow along) the sections that satisfy the inequality — arrows pointing outwards for "or" solutions, a segment between the circles for "and" solutions.

Is this topic on Foundation tier?

Quadratic inequalities are a Higher tier topic in all major GCSE specifications. Foundation tier focuses on linear inequalities only. If you are sitting Foundation GCSE, you need to know how to solve and represent linear inequalities (e.g. 2x + 1 < 7), but you will not be asked for quadratic inequalities.

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