Practice Quadratic equations
A quadratic equation contains a squared variable and may have zero, one or two real solutions.
Factorisation, completing the square and solution formulas offer different approaches.
Exercises with solutions
First rearrange the equation into the form x² + px + q = 0, then substitute p and q into the quadratic formula. The discriminant under the root tells you how many solutions exist; check each one by substitution.
Exercise 1
Solve x² − 5x + 6 = 0.
Show solution and working
x = 2 or x = 3
Here p = −5 and q = 6. The discriminant is 2.5² − 6 = 0.25, and its root is 0.5. So x = 2.5 ± 0.5, giving 2 and 3.
Exercise 2
Solve x² + 4x = 0.
Show solution and working
x = 0 or x = −4
Factor out x: x(x + 4) = 0. A product is zero exactly when one factor is zero, so x = 0 or x + 4 = 0.
Exercise 3
How many real solutions does x² + 2x + 5 = 0 have?
Show solution and working
no real solution
Here p = 2 and q = 5. The discriminant is 1² − 5 = −4. It is negative, so there is no real solution. The term (x + 1)² + 4 is always at least 4.
Exercise 4
Solve 2x² − 8 = 0.
Show solution and working
x = 2 or x = −2
Add 8 and divide by 2: x² = 4. Taking the root gives two solutions, because (−2)² = 4 as well.
Common mistake
Giving only the positive root. From x² = 4 there are two solutions, not just 2. Also make sure the equation is in standard form first, otherwise p and q are wrong.
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