Completing the Square

In one sentence

x^{2} + b x = (x + \frac{b}{2})^{2} - (\frac{b}{2})^{2}

A square with a strip stuck to its side is just one corner short of a perfect square. Fill in the corner, then take it away again: the expression has not changed, but now it is “a square plus a constant”, and a square is never less than $0$ . Everything that completing the square lets you do comes from this one fact.

Example: complete the square for x² + 6x + 5

Step 1 / 5

Ignore the constant for now
Focus on $x^{2} + 6 x$ : a square of side $x$ plus a strip measuring $6 \times x$ .

$x^{2} + 6 x + 5$
Cut the strip in half
Move one half underneath
Fill the corner, then pay it back
Tidy up the constants

Stop 1 · Solving equations (Vol. 2)

Completing the square turns “quadratic” into “take a square root”.

Predict first

How many real solutions does $(x + 3)^{2} = 4$ have?

1

2

None

Choose an answer first; then you can see the result.

One problem, several solutions

Solve $x^{2} + 6 x + 5 = 0$ .

Cross method

$5 = 1 \times 5$ and $1 + 5 = 6$ , so $(x + 1) (x + 5) = 0$ , giving $x = - 1$ or $x = - 5$ .

CostFastest, but only if whole numbers fit.

$(x + 3)^{2} - 4 = 0$ , so $(x + 3)^{2} = 4$ and $x + 3 = \pm 2$ , giving $x = - 1$ or $x = - 5$ .

CostTwo more lines, but it works whether or not the roots are whole numbers.

Stop 2 · Finding a minimum (Vol. 3)

After completing the square, $y = (x - h)^{2} + k$ . The square is at least $0$ , so the smallest value of $y$ is $k$ , reached when $x = h$ . The vertex can be read straight off the expression.

Drag

b

and

c

. Is the minimum reading always the same number as the constant at the end of the expression?

Stop 3 · Add a term, then factor (Vol. 1)

$x^{4} + 4$ looks impossible to factor: it has no real roots. But it is only one middle term away from a perfect square.

Example: factor x⁴ + 4

Step 1 / 4

See it as two squares
$x^{4} = (x^{2})^{2}$ and $4 = 2^{2}$ . The perfect square $(x^{2} + 2)^{2}$ would also need a middle term $2 \cdot x^{2} \cdot 2 = 4 x^{2}$ .

$(x^{2})^{2} + 2^{2}$
Something from nothing
Make the first three terms a square
Difference of squares

Stop 4 · Never negative (Vol. 2)

If several squares add up to $0$ , there is only one possibility: each of them is $0$ . That is how a single equation can pin down two unknowns.

Suppose $x^{2} + y^{2} - 4 x + 6 y + 13 = 0$ . Complete the square in $x$ and in $y$ separately, splitting $13$ as $4 + 9$ :

(x^{2} - 4 x + 4) + (y^{2} + 6 y + 9) = 0 ⟹ (x - 2)^{2} + (y + 3)^{2} = 0

So $x = 2$ and $y = - 3$ .

Practice

Exercise 1Basic

What is the minimum value of $x^{2} - 8 x + 9$ ?

Exercise 2Advanced

Given $a^{2} + b^{2} - 2 a + 6 b + 10 = 0$ , find $a$ and $b$ . (Enter both numbers separated by a comma, in any order.)

Mistake Museum · Exhibits from this stop

Before you leave, look at the traps other people fell into here.See every exhibit in the museum →

Exhibit

Borrowing without paying back

x^{2} + 6 x + 5 = (x + 3)^{2} + 5

What went wrong? Think first, then open

Diagnosis $(x + 3)^{2}$ expands to $x^{2} + 6 x + 9$ , which is $9$ more than the original $x^{2} + 6 x$ . The corner you filled in was borrowed; if you never subtract it again, the value of the expression changes.

x^{2} + 6 x + 5 = (x + 3)^{2} - 9 + 5 = (x + 3)^{2} - 4

“Add a term, and you must take it away again”

Exhibit

Not taking the leading coefficient out properly

2 x^{2} + 8 x = 2 (x^{2} + 4 x) = 2 (x + 2)^{2} - 4

What went wrong? Think first, then open

Diagnosis The $4$ added inside the brackets is doubled by the $2$ outside: $2 (x + 2)^{2} = 2 x^{2} + 8 x + 8$ , so the extra amount is really $8$ , not $4$ .

2 x^{2} + 8 x = 2 [(x + 2)^{2} - 4] = 2 (x + 2)^{2} - 8

. Borrow and pay back inside the brackets first, then multiply out.

“What you borrow inside the brackets, pay back inside the brackets”

Completing the Square

In one sentence

Stop 1 · Solving equations (Vol. 2)

Cross method

Completing the square

Stop 2 · Finding a minimum (Vol. 3)

Stop 3 · Add a term, then factor (Vol. 1)

Stop 4 · Never negative (Vol. 2)

Practice

Mistake Museum · Exhibits from this stop

Stops on this line