Rewrite x² + bx + c as (x + p)² + q to find turning points, exact solutions and minimum values.
What you will learn
- Write x² + bx + c as (x + p)² + q
- Read the turning point from it
- Solve by completing the square
Halve the x number
(x + 3)² = x² + 6x + 9.
So x² + 6x is (x + 3)² with 9 too many.
- x² + 6x + 2
- halve 6: (x + 3)² = x² + 6x + 9
- x² + 6x + 2 = (x + 3)² − 9 + 2
- = (x + 3)² − 7
The number in the bracket is always half the coefficient of x. Then take away its square and add the number term. An odd coefficient gives a fraction: x² + 3x + 1 = (x + 3/2)² − 9/4 + 1 = (x + 3/2)² − 5/4.
The rest of this lesson and its 7 practice questions are locked.
In this lesson
- Halve the x number
- The turning point for free
- Solving with it
- Try it: Find the lowest point
7 practice questions, marked as you go; about 25 minutes in all.
In this unit: Quadratics
Expand and factorise quadratics, solve them by factorising, the formula and completing the square, and read roots and turning points from graphs.