Completing the Square (Ex 5I)

Year 10 Mathematics Core · Cambridge Ch 5, §5I · Mr Wong

Name

Class

Date

Learning intentions.

Complete the square to factorise when integers can't be found
Solve quadratic equations by completing the square (surd answers)
Recognise a quadratic that gives no real solution

Key results

For $x^2+bx$, add and subtract $\left(\tfrac b2\right)^2$ to make $\left(x+\tfrac b2\right)^2$. Then $(x+p)^2=q$ gives $x=-p\pm\sqrt q$. If $q<0$ there is no real solution.

Warm-up — what number completes the square?

1. State the number to add to $x^2+bx$ to form a perfect square.

a $x^2+6x$

b $x^2-10x$

c $x^2+14x$

d $x^2-3x$

Part A — Completed-square form

2. Write each in the form $(x+p)^2+q$.

a $x^2-2x-4$

b $x^2+8x+3$

Part B — Solving (whole-number $b$)

3. Solve by completing the square. Leave surds exact.

a $x^2-2x-4=0$

b $x^2+8x+3=0$

c $x^2-6x+4=0$

d $x^2+2x-1=0$

5I — continued

Part C (fractional $b$) · Part D (no solutions) · Challenge

Name

Part C — Fractional middle term

4. Solve $x^2-x-1=0$ by completing the square. Give the answer as a single fraction.

5. Solve $x^2-5x+2=0$ by completing the square. Leave the surd exact.

Part D — How many solutions?

6. For each equation, complete the square and state how many real solutions it has.

a $(x-1)^2+7=0$

b $x^2+5=0$

Challenge (Extension, $a\ne 1$)

7. Solve $3x^2-12x=15$ by first dividing through by $3$, then completing the square.