Sheet № 89 · Higher only · AQA · Edexcel · OCR
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.
§Key definitions
Question:
Solve x² + 2x - 5 = 0, giving your answers in surd form.
Answer:
x = -1 + sqrt(6) or x = -1 - sqrt(6)
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.
§Formulas to memorise
x = (-b ± sqrt(b² - 4ac)) / 2a
Discriminant = b² - 4ac
Rearrange the equation into the standard form ax² + bx + c = 0.
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.)
a = 2, b = 3, c = 4.
Discriminant = b² - 4ac = 9 - 4(2)(4) = 9 - 32 = -23.
Worked example
Solve x² + 2x - 5 = 0, giving your answers in surd form.
Working:
⚠ 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.