Year 10 Mathematics Core — Solving by Completing the Square

Cambridge Ch 5 — Section 5I  •  Sun 14 June 2026
📚 Also for this topic: 📄 Printable worksheet ✅ Solutions (answer key)

Today's lesson

Not every quadratic factorises with whole numbers. When you can't find integers that work, completing the square lets you solve it anyway — often with a surd in the answer.

Learning intentions

Part 1 — Where does $\sqrt6$ come in? (~6 min)

Consider $x^2-2x-5=0$. There are no integers that multiply to $-5$ and add to $-2$, so it won't factorise nicely. Completing the square fixes that.

$x^2-2x-5=0 \;\Rightarrow\; x^2-2x+1-1-5=0 \;\Rightarrow\; (x-1)^2-6=0$
$(x-1)^2=6 \;\Rightarrow\; x-1=\pm\sqrt6 \;\Rightarrow\; x=1\pm\sqrt6$

Key idea — completing the square

Building understanding — what number completes the square for $x^2+bx$? (Add $\left(\tfrac b2\right)^2$.)

  1. $x^2+2x$
  2. $x^2+20x$
  3. $x^2-4x$
  4. $x^2+5x$
a) add $1$ → $(x+1)^2$    b) add $100$ → $(x+10)^2$    c) add $4$ → $(x-2)^2$    d) add $\tfrac{25}{4}$ → $\left(x+\tfrac52\right)^2$

Part 2 — Solving by completing the square (Example 1, ~16 min)

EXAMPLE 1 — Surd answers
Solve by first completing the square: a $x^2-4x+2=0$   b $x^2+6x-11=0$.
a   $x^2-4x+2=0$   (half of $-4$ is $-2$, and $(-2)^2=4$)
$x^2-4x+4-4+2=0 \;\Rightarrow\; (x-2)^2-2=0 \;\Rightarrow\; (x-2)^2=2$
$x-2=\pm\sqrt2 \;\Rightarrow\; \boxed{x=2\pm\sqrt2}$
b   $x^2+6x-11=0$   (half of $6$ is $3$, and $3^2=9$)
$x^2+6x+9-9-11=0 \;\Rightarrow\; (x+3)^2-20=0 \;\Rightarrow\; (x+3)^2=20$
$x+3=\pm\sqrt{20}=\pm 2\sqrt5 \;\Rightarrow\; \boxed{x=-3\pm 2\sqrt5}$

Now you try: Solve $x^2-6x+2=0$   and   $x^2+4x-14=0$.   Answers: $x=3\pm\sqrt7$;   $x=-2\pm 3\sqrt2$.

📺 Walkthrough: solving $x^2-4x+2=0$ by completing the square — add and subtract $\left(\tfrac b2\right)^2$, then take the square root for the surd answer.

Part 3 — A fractional case & no solutions (Example 2, ~12 min)

EXAMPLE 2 — Fractions and the no-solution case
Solve: a $x^2-3x+1=0$   b $x^2+5=0$.
a   $x^2-3x+1=0$   (half of $-3$ is $-\tfrac32$, and $\left(\tfrac32\right)^2=\tfrac94$)
$x^2-3x+\tfrac94-\tfrac94+1=0 \;\Rightarrow\; \left(x-\tfrac32\right)^2-\tfrac54=0$
$\left(x-\tfrac32\right)^2=\tfrac54 \;\Rightarrow\; x-\tfrac32=\pm\dfrac{\sqrt5}{2} \;\Rightarrow\; \boxed{x=\dfrac{3\pm\sqrt5}{2}}$
b   $x^2+5=0 \;\Rightarrow\; x^2=-5$. A square can't be negative, so there is no real solution.

Now you try: Solve $x^2-5x+2=0$.   Answer: $x=\dfrac{5\pm\sqrt{17}}{2}$.   And explain why $(x-1)^2+7=0$ has no real solution.

Practice 3.1 — solve by completing the square (leave surds exact). One has no real solution.

  1. $x^2-2x-4=0$
  2. $x^2+8x+3=0$
  3. $x^2-6x+4=0$
  4. $(x-1)^2+7=0$
a) $(x-1)^2=5$, so $x=1\pm\sqrt5$
b) $(x+4)^2=13$, so $x=-4\pm\sqrt{13}$
c) $(x-3)^2=5$, so $x=3\pm\sqrt5$
d) $(x-1)^2=-7$ → no real solution

Part 4 — Quick quiz (5 min)

Pick the correct answer for each, then click Mark.

Q1. To complete the square for $x^2+8x$, you add:

Q2. $x^2-4x+2=0$ becomes:

Q3. The solutions of $(x-2)^2=2$ are:

Q4. How many real solutions does $x^2+5=0$ have?

Q5. $\sqrt{20}$ written in simplest surd form is:

Working program — Cambridge Ex 5I (p462)

After the quiz, open Cambridge Chapter 5 (page 462) and complete the following:

Set workExtension
Questions 1–4 (2nd column), 5, 6a,c,e, 7a,c,e, 8, 9, 10Questions 11, 12

Leave surd answers exact and in simplest form (e.g. $2\sqrt5$, not $\sqrt{20}$).

Exit ticket — write in your book

Before you pack up, write one sentence each:
  1. What number completes the square for $x^2+10x$?
  2. Why might a quadratic answer contain a surd?
  3. Why does $x^2+5=0$ have no real solution?

— Mr Wong