Completing the square

Factorisation works only when the quadratic splits over rationals. Completing the square is a method that always works , for any quadratic, even those with irrational or no real roots.

The idea

The identity $(x + h)^2 = x^2 + 2 h x + h^2$ tells us that any expression of the form $x^2 + 2 h x$ is almost a perfect square; it just needs $+ h^2$ to complete it.

So to complete the square of $x^2 + 2 h x + k$, we write $x^2 + 2 h x + k = (x^2 + 2 h x + h^2) + (k - h^2) = (x + h)^2 + (k - h^2).$

Now any quadratic equation $x^2 + 2 h x + k = 0$ becomes $(x + h)^2 = h^2 - k,$ and we can square-root both sides (when the RHS is non-negative) to find $x$.

Algorithm

To solve $a x^2 + b x + c = 0$ by completing the square:

1. Divide through by $a$: $x^2 + \dfrac{b}{a} x + \dfrac{c}{a} = 0$.
2. Move the constant: $x^2 + \dfrac{b}{a} x = -\dfrac{c}{a}$.
3. Add $\left(\dfrac{b}{2 a}\right)^2$ to both sides: $x^2 + \frac{b}{a} x + \left(\frac{b}{2 a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4 a^2}.$
4. Rewrite the LHS as a square: $\left(x + \frac{b}{2 a}\right)^2 = \frac{b^2 - 4 a c}{4 a^2}.$
5. Take square roots and solve for $x$: $x + \frac{b}{2 a} = \pm \frac{\sqrt{b^2 - 4 a c}}{2 a} \implies x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}.$

This is the quadratic formula.

Deriving the formula

Notice how completing the square produces the formula. The quantity under the square root is exactly the discriminant $D = b^2 - 4 a c$. The formula tells us:

• if $D > 0$: two distinct real roots,
• if $D = 0$: one repeated real root $x = -b/(2 a)$,
• if $D < 0$: no real root (the square root of a negative number is not real).

So one derivation gives us both how to solve and what kind of solutions to expect.

Worked examples

Example 1. Solve $x^2 + 6 x + 7 = 0$ by completing the square.

$x^2 + 6 x = -7 \Rightarrow x^2 + 6 x + 9 = 2 \Rightarrow (x + 3)^2 = 2 \Rightarrow x + 3 = \pm \sqrt{2} \Rightarrow x = -3 \pm \sqrt{2}$.

Example 2. Solve $2 x^2 - 7 x + 3 = 0$ by completing the square.

Divide by $2$: $x^2 - (7/2) x + 3/2 = 0$. Move constant: $x^2 - (7/2) x = -3/2$.

Add $(7/4)^2 = 49/16$: $x^2 - (7/2) x + 49/16 = -3/2 + 49/16 = -24/16 + 49/16 = 25/16$.

$\left(x - 7/4\right)^2 = 25/16 \Rightarrow x - 7/4 = \pm 5/4 \Rightarrow x = (7 \pm 5)/4 = 3, 1/2$.

Example 3. Solve $4 x^2 + 4 x - 3 = 0$ by completing the square.

Divide by $4$: $x^2 + x - 3/4 = 0$. Move constant: $x^2 + x = 3/4$.

Add $(1/2)^2 = 1/4$: $x^2 + x + 1/4 = 1$. $\left(x + 1/2\right)^2 = 1$. $x + 1/2 = \pm 1$. $x = 1/2, -3/2$.

Example 4. Solve $x^2 + 4 x + 8 = 0$ by completing the square.

$x^2 + 4 x = -8 \Rightarrow (x + 2)^2 = 4 - 8 = -4$. Square of a real number cannot be negative , no real solutions.

(This matches $D = 16 - 32 = -16 < 0$.)

Example 5. Solve $x^2 - 4 x + 4 = 0$.

$x^2 - 4 x + 4 = (x - 2)^2 = 0 \Rightarrow x = 2$ (repeated).

Try it yourself

1. Solve $x^2 - 4 x - 5 = 0$ by completing the square.
2. Solve $x^2 + 8 x + 12 = 0$ by completing the square.
3. Solve $3 x^2 + 5 x + 2 = 0$ by completing the square.
4. Solve $2 x^2 - 4 x - 3 = 0$ by completing the square.
5. Solve $x^2 + 2 x + 2 = 0$. Does the equation have real solutions?
6. Solve $5 x^2 + 6 x = 4$ by completing the square.
7. Solve $x^2 + x - 1 = 0$ by completing the square; compare with the quadratic formula.
8. Show that $x^2 + 2 x + 3 > 0$ for all real $x$ by completing the square.
9. Solve $x^2 - (\sqrt{3} + 1) x + \sqrt{3} = 0$ by completing the square.
10. Solve $9 x^2 - 15 x + 6 = 0$ by completing the square.

Pitfalls / Insight

• Always divide by the leading coefficient first. Trying to complete the square with $a \ne 1$ at the front leads to arithmetic errors.
• Don't forget to add $(b/2)^2$ to BOTH sides , completing the square is symmetric.
• Recognise $(x + h)^2 = \text{negative}$ as a signal of no real solutions, not as a mistake.

Insight. Completing the square is the derivation of the quadratic formula , and you'll see the same move (turning into a sum of squares) in coordinate geometry, calculus and statistics later. Master it once and reuse.