Quadratic equations · Grade 11
In Grade 11 you learn: quadratics from problems.
A quadratic equation has an x² term, like x² + 5x + 6 = 0, and usually has two solutions. You can solve it by factorising, by completing the square, or with the quadratic formula, which always works.
Key facts
- Rearrange so that one side is 0 before you solve.
- If two brackets multiply to 0, one of them must be 0: (x − 2)(x + 5) = 0 gives x = 2 or x = −5.
- Quadratic formula for ax² + bx + c = 0: x = (−b ± √(b² − 4ac))/(2a).
- The discriminant b² − 4ac: positive → two solutions; zero → one repeated solution; negative → no real solutions.
- x² = 25 has two solutions, x = 5 and x = −5.
- The solutions are where the graph of y = ax² + bx + c crosses the x-axis.
Worked examples
Example 1
Solve x² + 4x − 7 = 0, giving your answers to 2 decimal places.
Answer: x = 1.32 or x = −5.32