Completing the square GCSE lesson: turning points and solving quadratics

Completing the square: practice

Show your working, and expand your square back out to check it. The stretch is optional.

Diagnostic question: choose and explain

1. x² − 6x + 2 = (x − 3)² + q. What is q?
   1. −7
   2. 11
   3. −1
   4. 2

Practice questions

1. x² + 10x + 7 = (x + ___)² − ___
2. x² − 8x + 20 = (x − ___)² + ___
3. The lowest point of y = x² + 2x − 3 is (___, ___)
4. Solve x² − 2x − 4 = 0 by completing the square. Give the larger solution to 2 decimal places.
5. y = (x − 3)² + 5. How many times does the curve meet the x-axis?
   1. Never
   2. Once
   3. Twice
6. Which curve has its lowest point at (−2, 3)?
   1. y = (x + 2)² + 3
   2. y = (x − 2)² + 3
   3. y = (x + 3)² − 2
7. x² + bx + 14 = (x + 3)² + c, so b = ___ and c = ___
8. Stretch: 2x² + 12x + 7 = 2(x + ___)² − ___

Support: For question 1: half of 10 is 5, so start with (x + 5)². It expands to x² + 10x + 25. You only want + 7: what must you take away?

Completing the square: exit

Show your working; one sentence for the reason.

1. Write x² − 10x + 30 in the form (x − p)² + q. Does y = x² − 10x + 30 meet the x-axis? Say how you know.

Exit check

One square completed and one reason show who links (p, q) to the roots.

Write x² − 10x + 30 in the form (x − p)² + q.

Completing the square: turning points and solving

Show each square completed, and expand it back out to check. Give exact answers with surds for question 5.