A quadratic equation in one variable has the form ax² + bx + c = 0 with a ≠ 0. Solving it means finding the values of x that make the equation true. Always put the equation into this standard form before identifying its coefficients or choosing a method.
Factorise when the numbers cooperate
Factorisation rewrites a quadratic expression as a product. If two factors multiply to zero, at least one factor is zero. For x² − 7x + 12, look for two numbers whose product is 12 and whose sum is −7. They are −3 and −4.
For a coefficient of x² other than 1, you can split the middle term using a suitable pair with product ac and sum b. Check the expansion of your factors before solving the smaller equations.
- A repeated factor can give a repeated root.
- Do not divide by x unless you have separately considered x = 0.
Formula and discriminant
For ax² + bx + c = 0, the quadratic formula is x = (−b ± √(b² − 4ac)) / (2a). The denominator 2a applies to the whole numerator. Keep negative coefficients in brackets when substituting.
The discriminant D = b² − 4ac describes the real roots: D > 0 gives two distinct real roots; D = 0 gives equal real roots; D < 0 gives no real roots. It does not tell you that every root must be an integer.
Put it into practice
A worked example
Solve x² − 7x + 12 = 0.
- Write x² − 3x − 4x + 12 = 0.
- Factor to obtain (x − 3)(x − 4) = 0.
- Set each factor equal to zero and substitute back to check.
Answer: x = 3 or x = 4. Both satisfy the original equation.
Try it yourself
Have a go before revealing the answer.
Solve 2x² − 5x + 2 = 0.
Show answer
(2x − 1)(x − 2) = 0, so x = 1/2 or 2.
Answer: (2x − 1)(x − 2) = 0, so x = 1/2 or 2.
How many real roots does x² + 4x + 5 = 0 have?
Show answer
D = 16 − 20 = −4, so it has no real roots.
Answer: D = 16 − 20 = −4, so it has no real roots.
Watch out for these
- The quadratic formula uses −b, not b.
- First rearrange both sides into a single expression equal to zero.
Continue with official resources
Match these concept notes to your current textbook and syllabus. They are independent explanations, not official board publications.