Quadratic Formula and Discriminant –

The quadratic formula and discriminant are essential Higher tier GCSE Maths topics. The formula solves any quadratic equation, while the discriminant tells you how many real solutions to expect before you even start solving.

What Are the Quadratic Formula and Discriminant?

The quadratic formula gives the solutions of any quadratic equation ax² + bx + c = 0:

x = (-b ± √(b² - 4ac)) / 2a

The expression under the square root, b² - 4ac, is called the discriminant. It determines the number of real solutions:

• If b² - 4ac > 0 (positive), there are two distinct real solutions.
• If b² - 4ac = 0, there is one repeated real solution (the parabola just touches the x-axis).
• If b² - 4ac < 0 (negative), there are no real solutions (the parabola does not cross the x-axis).

Understanding the discriminant allows you to answer "show that this equation has no solutions" or "find the values of k for which the equation has equal roots" — both common exam questions.

Key Formulas

Step-by-Step Method

1. Write the equation in the form ax² + bx + c = 0 and identify a, b, and c.
2. Calculate the discriminant: b² - 4ac.
3. If the discriminant is negative, state there are no real solutions and stop.
4. Substitute a, b, and the discriminant into the quadratic formula.
5. Calculate the two solutions using + and - for the ± symbol. Round if required.

Worked Example 1 — Foundation Level

Question: This is a Higher only topic. Here is a straightforward example. Solve x² + 5x + 6 = 0 using the quadratic formula.

Working:

Step 1 — Identify: a = 1, b = 5, c = 6.

Step 2 — Discriminant: 5² - 4(1)(6) = 25 - 24 = 1.

Step 3 — Apply the formula: x = (-5 ± √1) / (2 × 1) = (-5 ± 1) / 2.

Step 4 — Two solutions: x = (-5 + 1) / 2 = -4/2 = -2 and x = (-5 - 1) / 2 = -6/2 = -3.

Answer: x = -2 or x = -3

Worked Example 2 — Higher Level

Question: Solve 2x² - 3x - 4 = 0, giving your answers to 2 decimal places.

Working:

Step 1 — a = 2, b = -3, c = -4.

Step 2 — Discriminant: (-3)² - 4(2)(-4) = 9 + 32 = 41.

Step 3 — x = (3 ± √41) / 4.

Step 4 — √41 = 6.4031...

Step 5 — x = (3 + 6.4031) / 4 = 9.4031 / 4 = 2.35 (2 d.p.) and x = (3 - 6.4031) / 4 = -3.4031 / 4 = -0.85 (2 d.p.).

Answer: x = 2.35 or x = -0.85

Worked Example 3 — Exam Style

Question: The equation kx² + 6x + k = 0 has equal roots. Find the possible values of k. (3 marks)

Working:

Step 1 — For equal roots, the discriminant must equal zero: b² - 4ac = 0.

Step 2 — a = k, b = 6, c = k. So 6² - 4(k)(k) = 0.

Step 3 — 36 - 4k² = 0.

Step 4 — 4k² = 36, so k² = 9, giving k = 3 or k = -3.

Answer: k = 3 or k = -3

Common Mistakes

• Using the wrong sign for b. In the formula, the first term is -b. If b = -3, then -b = 3 (positive). Students often keep the negative sign, making -b = -3.
• Forgetting to divide the entire numerator by 2a. The division applies to both -b and the ±√(b² - 4ac) part, not just the square root.
• Not writing the equation as = 0 before identifying a, b, c. If the equation is x² + 3x = 7, you must rewrite as x² + 3x - 7 = 0 first.

Exam Tips

• If the question asks for answers to a given number of decimal places, the quadratic formula is usually the intended method (not factorising).
• Calculate the discriminant first and write it down — you earn a method mark even if you make an error in the final calculation.
• When asked "how many solutions," you only need the discriminant — you do not need to solve the equation fully.

Practice Questions

Q1 (Higher): Calculate the discriminant of 3x² + 2x - 5 = 0 and state the number of solutions.

Q2 (Higher): Solve x² - 6x + 2 = 0, giving answers to 2 decimal places.

Q3 (Higher): The equation x² + px + 9 = 0 has no real solutions. Find the range of values of p.

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Related Topics

• Solving Quadratic Equations: Factorising
• Completing the Square
• Quadratic Graphs

Summary

• The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) solves any quadratic equation.
• The discriminant b² - 4ac tells you how many real solutions exist: positive means two, zero means one, negative means none.
• Always rewrite the equation as ax² + bx + c = 0 before identifying a, b, and c.
• Watch the sign of b carefully — -b means change its sign.
• Calculate the discriminant first to earn method marks and to check your work.