Completing the Square

Quadratic expressions, functions, and equations can all be manipulated in useful ways with a method called completing the square. This method is based on the concept of perfect square trinomials.

Concept

Perfect Square Trinomial

Just as a perfect square integer, such as 9, can be written as the square of integer square roots, a perfect square trinomial is a trinomial which can be written as the square of a binomial.

There are two general types of perfect square trinomials that can be useful when dealing with quadratic functions or quadratic equations. This process involves factoring the quadratic trinomial and rewriting the perfect square trinomial as the square of a binomial.

Method

Factoring a Perfect Square Trinomial

For a trinomial to be factorable as a perfect square trinomial, the first and last terms must be perfect squares and the middle term must be two times the square roots of the first and last terms. Consider the following expression.

To factor this trinomial, there three steps.

1

Examine the First and Last Terms

One good way to recognize if a trinomial is perfect square is to look at its first and last term. If they are both perfect squares, there's a good chance that it may be a perfect square trinomial. In the given expression, first and last terms can be written as the squares of 4x and 11, respectively.

16 x^{2} + 88 x + 121 ⇓ (4 x)^{2} + 88 x + (11)^{2}

These perfect squares show that the expression can be a perfect square trinomial. However, this is not enough to decide yet.

2

Examine the Middle Term

The next step is to check whether the middle term is two times the square roots of the first and last terms.

16 x^{2} + 88 x + 121 ⇓ (4 x)^{2} + 2 (4 x) (11) + (11)^{2}

It can be seen that the given expression satisfies this condition, as well.

3

Write as a Square of a Binomial

Since the expression satisfies both conditions, it is a perfect square trinomial. Therefore, it can be written as a square of a binomial where $4 x$ and $11$ are the first and second terms of the binomial, respectively.

16 x^{2} + 88 x + 121 = (4 x + 11)^{2}

Create the perfect square trinomial

fullscreen

Exercise

The expressions in the table are perfect square trinomials. Complete the expressions to ensure equivalence between corresponding standard forms and factored forms.

standard form	factored form
x2−2x+1	(x__ ___)2
x2−___x+___	(x−7)2
x2−10x+___	(x__ ___)2
x2+___x+100	(x__ ___)2

Show Solution

Solution

Here the same expressions are written both in standard form and in factored form, but one or more terms are missing in each expression. To identify the missing term(s) the rules for factoring a perfect square trinomial are useful.

a2+2ab+b2=(a+b)2
and
a2−2ab+b2=(a−b)2

To help us identify the terms in the rule in the first example, we can for the expression x2−2x+1 rewrite 1 as 12 and 2x as 2⋅x⋅1.

a^{2} - 2 \cdot a \cdot b + b^{2} = (a \cdot - \cdot b)^{2}

x2−2⋅x⋅1+12=(x__ ___)2

To make the expressions match each other we will fill in −1 where the blanks are.

a^{2} - 2 \cdot a \cdot b + b^{2} = (a - b)^{2}

x2−2⋅x⋅1+12=(x−1)2

We can use this reasoning to complete the other perfect square trinomials in the table. Next, we'll consider the second row.

a2−2ab+b2=(a−b)2
x2−___x+___ =(x−7)2

To fill in the second blank we use that b must be 7, which gives us that b2=72. When we study the first blank we can match a with x. What is left is 2⋅b=2⋅7.

a^{2} - 2 \cdot a \cdot b + b^{2} = (a - b)^{2}

x2−2⋅x⋅7+72=(x−7)2

This we prefer writing as x2−14x+49=(x−7)2. Let's continue with the third row

a^{2} - 2 a b + b^{2} = (a . - . b)^{2}

x2−10x+___=(x__ ___)2

Here the terms a and x match each other. Since the terms 2ab and 10x must match each other we find that 2b=10, or b=5. Let's use this to fill in the blanks in the expression.

a^{2} - 2 a b + b^{2} = (a . - . b)^{2}

x2−10x+52=(x−5)2

By squaring 5 we get that the second row reads x2−10x+25=(x−5)2. We are now going to deal with the last row in the table.

a^{2} + 2 . a . b + b^{2} = (a . + . b)^{2}

x2+___x+100=(x__ ___)2

The lines match each other when b=10, giving us the first blank 2⋅10=20 and the second blank +10. Let's write them together with the blanks filled in.

a^{2} + 2 . a . b + b^{2} = (a . + . b)^{2}

x2+20x+100=(x+10)2

We'll summarize our results by adding the found values to the table.

standard form	factored form
x2−2x+1	(x−1)2
x2−14x+49	(x−7)2
x2−10x+25	(x−5)2
x2+20x+100	(x+10)2

Concept

Completing the Square

Completing the square is a method by which a quadratic expression is rewritten as a difference of a perfect square trinomial and a constant. Commonly this is done by adding and subtracting a constant, p.

By choosing the constant, p, as

(\frac{b}{2})^{2}

the quadratic expression becomes a difference of a perfect square trinomial and a constant.

perfect square trinomial x^{2} + b x + (\frac{b}{2})^{2} - constant (\frac{b}{2})^{2}

The quadratic expression can then be factored and written and written in vertex form.

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

Extra

Completing the Square, In-Depth

Concept

Quadratic Expressions in the Form x2+bx+c

When completing the square for a quadratic expression in the form x2+bx+c, the constant c must also be taken into account. When adding and subtracting the term

(\frac{b}{2})^{2},

the quadratic expression gets a constant term of

- (\frac{b}{2})^{2} + c .

x^{2} + b x + c ⇓ perfect square trinomial x^{2} + b x + (\frac{b}{2})^{2} - constant (\frac{b}{2})^{2} + c

Concept

Factoring

After completing its square, a quadratic expression can be rewritten into vertex form by factoring the perfect square trinomial.

perfect square trinomial x^{2} + b x + (\frac{b}{2})^{2} - (\frac{b}{2})^{2} + c = (x + \frac{b}{2})^{2} - (\frac{b}{2})^{2} + c

Concept

Solving Quadratic Equations

Completing the square can be used when solving quadratic equations. An equation in the form x2+bx+c=0 is then rewritten by isolating the x2- and x-terms.

By adding the term

(\frac{b}{2})^{2}

to both sides of the equation, a perfect square trinomial is formed.

The equation can then be rewritten by factoring the perfect square trinomial.

When the equation is written on this form it can be solved with square roots.

Concept

Geometric Interpretation of Completing the Square

It can be helpful to visualize what's going on when completing the square. Press the button below to see an animation of this concept.

Complete the square

Method

Completing the Square

This is a method by which a quadratic expression is rewritten as the difference of the square of a binomial and a constant.

x^{2} \pm b x + c ⇓ square (x \pm \frac{b}{2})^{2} - constant (\frac{b}{2})^{2} + c

To complete the square, there are five steps to follow.

1

Factor Out the Coefficient of x2

It is easier to complete the square when the expression is written in the form x2±bx+c. Therefore, if the coefficient of x2 is not 1, it should be factored out.

For the simplicity of the following steps, it will be assumed that a=1. For values of a other than 1, the same steps should be performed. The only difference is that the new coefficients —

\frac{b}{a}

instead of b and

\frac{c}{a}

instead of c — will be used.

2

Identify the Constant Needed to Complete the Square

The constant needed to complete the square can now be identified by focusing on the x-term, while ignoring the rest. One way to find this constant is by squaring half the coefficient of the x-term, which in this case is b.

Note that leaving the constant as a power makes the next steps easier to perform.

3

Complete the Square

The square can now be completed by adding and subtracting the constant found in Step 2. Note that the value of the original expression will not be changed since the result of adding and subtracting the same value is equal to 0.

x^{2} \pm b x + c ⇕ perfect square trinomial x^{2} \pm b x + (\frac{b}{2})^{2} - constant (\frac{b}{2})^{2} + c

The first three terms form a perfect square trinomial, which can be factored as the square of a binomial. The other two terms do not contain the variable x, so their value is constant.

4

Factor the Perfect Square Trinomial

The perfect square trinomial can now be factored and rewritten as the square of a binomial.

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

DenomMultFracToNumber

$2 \cdot \frac{a}{2} = a$

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

CommutativePropMult

Commutative Property of Multiplication

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

a2±2ab+b2=(a±b)2

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

The process of completing the square is now finished.

5

Simplify the expression

Finally, if needed, the expression can be simplified. In case a was not equal to 1, now is a good time to remove the parentheses and multiply the obtained expression by a.