Completing the square · Grade 10
In Grade 10 you learn: turning points from completed square form.
Completing the square rewrites a quadratic such as x² + 6x + 2 as (x + 3)² − 7. This form shows the turning point of the graph straight away and lets you solve any quadratic — it is where the quadratic formula comes from.
Key facts
- x² + bx + c = (x + b/2)² − (b/2)² + c.
- Halve the coefficient of x to get the number in the bracket, then subtract its square.
- (x + p)² + q has minimum value q when x = −p, so its turning point is (−p, q).
- If x² has a coefficient, take it out of the first two terms first: 2x² + 12x + 5 = 2(x + 3)² − 13.
- To solve, write it as (x + p)² = k, then square root both sides (remember ±).
Worked examples
Example 1
Find the turning point of y = x² − 6x + 13.
Answer: (3, 4), a minimum