Find a minimum
Complete the square for x² + 6x + 5.
- Half of 6 is 3; (x + 3)² expands to x² + 6x + 9.
- Subtract 9 and add 5, leaving (x + 3)² − 4.
- The square is least at x = −3.
Answer: (x + 3)² − 4; vertex (−3, −4)
Quadratics: roots and turning points
Use factorised form to find roots and completed-square form to find the turning point.
Before you start
You’ll need Expanding and factorising. Follow a link if you want to revise it first.
Reading a quadratic in different forms
In (x − a)(x − b), the roots are a and b. In (x − h)² + k, the turning point is (h, k). Expanding either expression gives the same polynomial.
For x² + bx, add and subtract (b/2)². The added square must be balanced by an equal subtraction.
A vertex is not automatically a root. The minimum or maximum is the y value at the vertex.
Both (x − 2)² + 1 and (x − 2)² − 1 have a turning point at x = 2. The first has no real roots; the second has two.
A minimum and two roots
Recover roots
Solve x² − 5x + 6 = 0.
- Find two numbers with sum −5 and product 6: −2 and −3.
- Factor as (x − 2)(x − 3) = 0.
- At least one factor must be zero.
Answer: x = 2 or x = 3
The sign in the bracket is opposite to the vertex x coordinate.
Practice questions
The first link uses smaller values. Move on when you can answer without referring to the examples. The difficulty settings aren’t tied to exam grades.
Practice is filtered to your selected course. Change course or options when needed.
Check your understanding
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Which form would you choose to find roots, and which to find a turning point?