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MathidiaMathematics Academy
Quadratic equations and graphs
Lesson 18 of 75
Standard practice

The quadratic formula

Objective. Use the formula accurately and interpret discriminant information.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The quadratic formula solves any equation of the form ax² + bx + c = 0 using x = (−b ± √(b² − 4ac)) ÷ 2a, even when the expression will not factorise neatly. The discriminant b² − 4ac inside the root tells you the nature of the roots: positive gives two real solutions, zero gives one repeated solution, and negative gives no real solutions. Substitute a, b and c carefully, keeping track of negative signs, and simplify the surd if possible. Always write the two solutions separately using + and − before evaluating.

Worked example

Solve 2x² − 7x + 3 = 0 using the quadratic formula, giving exact solutions.

2 … 7
Illustrated card showing 2 … 7
  1. 1Identify a = 2, b = −7, c = 3.
  2. 2Calculate the discriminant: (−7)² − 4(2)(3) = 49 − 24 = 25.
  3. 3Apply the formula: x = (7 ± √25) ÷ 4 = (7 ± 5) ÷ 4.
  4. 4Evaluate both solutions: x = 3 or x = 1/2.

Answer: x = 3 or x = 1/2

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Common mistake. Pupils often substitute b incorrectly when b is negative, computing −b as negative instead of positive, e.g. for b = −5 writing −b = −5 rather than 5. Always flip the sign of b carefully when substituting.

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