Using the Quadratic Formula –

The quadratic formula is a universal method for solving any quadratic equation, even when it does not factorise neatly. Although the formula is printed on the exam formula sheet, knowing how to use it fluently under time pressure is essential for Higher tier success.

What Is the Quadratic Formula?

The quadratic formula solves equations of the form ax² + bx + c = 0 by substituting the coefficients a, b, and c directly into a formula. It always works, whether the solutions are integers, fractions, surds, or non-existent.

The part of the formula underneath the square root sign, b² - 4ac, is called the discriminant. It determines how many real solutions the equation has. If the discriminant is positive, there are two distinct solutions. If it equals zero, there is exactly one repeated solution. If it is negative, there are no real solutions — the parabola does not cross the x-axis.

Understanding the discriminant is valuable because exam questions sometimes ask "how many solutions does this equation have?" without requiring you to solve it.

Key Formulas

Step-by-Step Method

1. Rearrange the equation into the standard form ax² + bx + c = 0.
2. Identify the values of a, b, and c, paying careful attention to negative signs.
3. Calculate the discriminant b² - 4ac separately.
4. Substitute a, b, and the discriminant into the formula.
5. Evaluate the two solutions using + and - in the numerator.
6. Simplify surds or round to the required number of decimal places.

Worked Example 1 — Foundation Level

Question: Solve x² + 2x - 5 = 0, giving your answers in surd form.

Working:

Step 1 — Identify a = 1, b = 2, c = -5.

Step 2 — Discriminant: b² - 4ac = (2)² - 4(1)(-5) = 4 + 20 = 24.

Step 3 — Substitute into the formula: x = (-2 ± sqrt(24)) / 2(1) = (-2 ± sqrt(24)) / 2.

Step 4 — Simplify sqrt(24) = sqrt(4 times 6) = 2sqrt(6).

Step 5 — x = (-2 ± 2sqrt(6)) / 2 = -1 ± sqrt(6).

Answer: x = -1 + sqrt(6) or x = -1 - sqrt(6)

Worked Example 2 — Higher Level

Question: Solve 3x² - 5x - 1 = 0, giving your answers correct to 2 decimal places.

Working:

Step 1 — a = 3, b = -5, c = -1.

Step 2 — Discriminant: (-5)² - 4(3)(-1) = 25 + 12 = 37.

Step 3 — x = (-(-5) ± sqrt(37)) / 2(3) = (5 ± sqrt(37)) / 6.

Step 4 — sqrt(37) = 6.0828 (to 4 d.p.).

x = (5 + 6.0828) / 6 = 11.0828 / 6 = 1.847... = 1.85 (2 d.p.)

x = (5 - 6.0828) / 6 = -1.0828 / 6 = -0.1804... = -0.18 (2 d.p.)

Answer: x = 1.85 or x = -0.18

Worked Example 3 — Exam Style

Question: Use the discriminant to show that 2x² + 3x + 4 = 0 has no real solutions. (2 marks)

Working:

a = 2, b = 3, c = 4.

Discriminant = b² - 4ac = 9 - 4(2)(4) = 9 - 32 = -23.

Since the discriminant is negative (-23 < 0), the equation has no real solutions.

Answer: Discriminant = -23; no real solutions exist.

Common Mistakes

• Getting the sign of b wrong. The formula contains -b. If b is already negative (e.g. b = -5), then -b = -(-5) = +5. Write this substitution out explicitly to avoid errors.
• Dividing only part of the numerator by 2a. The entire expression -b ± sqrt(b² - 4ac) must be divided by 2a, not just one part. Draw a long fraction line under the whole numerator.
• Rounding too early. Keep the full square root value until the final step. Rounding intermediate calculations introduces error and can give an incorrect final answer.

Exam Tips

• Write "a = ..., b = ..., c = ..." on a separate line before substituting. This earns a method mark on most papers and helps you avoid sign mistakes.
• If the question asks for answers to a number of decimal places, that is a strong hint to use the quadratic formula rather than factorising.
• Calculate the discriminant on its own line. It keeps your working tidy and immediately tells you whether to expect two solutions, one, or none.

Practice Questions

Q1 (Higher): Solve x² - 6x + 4 = 0, giving answers in surd form.

Q2 (Higher): Solve 2x² + x - 4 = 0, giving answers to 2 decimal places.

Q3 (Higher): How many real solutions does 5x² - 2x + 1 = 0 have? Justify your answer.

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Related Topics

• Solving Quadratic Equations Quadratic Formula
• Completing the Square
• Solving Quadratic Equations by Factorising

Summary

• The quadratic formula x = (-b ± sqrt(b² - 4ac)) / 2a solves any quadratic equation in the form ax² + bx + c = 0.
• Identify a, b, and c carefully, especially when coefficients are negative.
• The discriminant b² - 4ac tells you how many real solutions exist: positive means two, zero means one, negative means none.
• Show full substitution in your working to earn method marks.
• Simplify surds when asked for exact answers; otherwise round only at the final step.