Perfect Square Trinomials

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A perfect square trinomial is a trinomial that equals a binomial squared. It fits a^2 + 2ab + b^2 = (a + b)^2 or a^2 - 2ab + b^2 = (a - b)^2. The first and last terms are perfect squares, and the middle term is 2 times their roots. For example, x^2 + 10x + 25 = (x + 5)^2.

Before this lesson: You should know special products of binomials, factoring trinomials and factoring the difference of squares.

What Is a Perfect Square Trinomial?

A trinomial is a polynomial with three terms. Some trinomials come from squaring a binomial. For example, (x + 5)^2 = x^2 + 10x + 25. Those are perfect square trinomials.

You saw this pattern in Special Products of Binomials. Now you run it backward. Look at the first and last terms, then check the middle term.

The trinomial 4x² + 12x + 9 with three checks. The first term 4x² is the square of 2x. The last term 9 is the square of 3. The middle term matches 2 · 2x · 3 = 12x. So 4x² + 12x + 9 = (2x + 3)².
The ends are squares of 2x and 3. The middle is 2 · 2x · 3 = 12x, so the trinomial is (2x + 3)².

Perfect Square Trinomial Formulas

\begin{aligned} a^2 + 2ab + b^2 &= (a + b)^2 \\ a^2 - 2ab + b^2 &= (a - b)^2 \end{aligned}

The sign of the middle term tells you the sign in the binomial. The last term is always positive, because it is a square.

Steps for factoring a perfect square trinomial

  1. Factor out the GCF, if there is one.
  2. Check that the first and last terms are perfect squares. The last term must be positive.
  3. Find a and b, the square roots of the first and last terms.
  4. Check that the middle term is 2ab. Ignore its sign for now.
  5. Write (a + b)^2 if the middle term is plus. Write (a - b)^2 if it is minus.
  6. If the middle term doesn’t match, it is not a perfect square. Factor another way.

Worked Examples

Example 1: A plus sign in the middle

Factor x^2 + 10x + 25.

The first term is x^2, so a = x. The last term is 25 = 5^2, so b = 5. The middle should be 2 \cdot x \cdot 5 = 10x. It is.

x^2 + 10x + 25 = (x + 5)^2

Example 2: A minus sign in the middle

Factor y^2 - 14y + 49.

Here a = y and b = 7. Check: 2 \cdot y \cdot 7 = 14y. The middle term is minus, so the binomial has a minus.

y^2 - 14y + 49 = (y - 7)^2

Check: (y - 7)^2 = y^2 - 7y - 7y + 49 = y^2 - 14y + 49.

Example 3: A coefficient on the first term

Factor 4x^2 + 12x + 9.

4x^2 = (2x)^2, so a = 2x. 9 = 3^2, so b = 3. The middle should be 2 \cdot 2x \cdot 3 = 12x. It matches.

4x^2 + 12x + 9 = (2x + 3)^2

Example 4: Factor out the GCF first

Factor 2x^2 - 24x + 72.

Every term is even, so take out 2. Then x^2 - 12x + 36 has a = x and b = 6, and 2 \cdot x \cdot 6 = 12x.

\begin{aligned} 2x^2 - 24x + 72 &= 2(x^2 - 12x + 36) \\ &= 2(x - 6)^2 \end{aligned}

Example 5: Word problem

A square rug has an area of 9x^2 + 12x + 4 square feet. Find the length of one side. Then find the side and the area when x = 2.

9x^2 = (3x)^2 and 4 = 2^2. The middle should be 2 \cdot 3x \cdot 2 = 12x. It matches, so the area is (3x + 2)^2.

9x^2 + 12x + 4 = (3x + 2)^2

Each side is 3x + 2 feet. When x = 2, a side is 3(2) + 2 = 8 feet, and the area is 8^2 = 64 square feet. Check: 9(4) + 12(2) + 4 = 64.

Common Mistakes

MistakeExample of the errorFix
Using the wrong signy^2 - 14y + 49 = (y + 7)^2A minus middle term gives a minus: (y - 7)^2.
Not checking the middle termx^2 + 13x + 36 = (x + 6)^22 \cdot x \cdot 6 = 12x, not 13x. Factor it another way: (x + 4)(x + 9).
Missing that the last term is negativex^2 - 6x - 9 = (x - 3)^2(x - 3)^2 = x^2 - 6x + 9. A negative last term is never a perfect square.
Not taking the root of the coefficient4x^2 + 12x + 9 = (4x + 3)^2The root of 4x^2 is 2x. The answer is (2x + 3)^2.
Mixing it up with a difference of squaresx^2 + 10x + 25 = (x + 5)(x - 5)(x + 5)(x - 5) = x^2 - 25. Three terms means (x + 5)^2.

How to Tell If a Trinomial Is a Perfect Square

TrinomialEnds are squares?Middle is 2ab?Perfect square?
x^2 + 6x + 9Yes: x and 3Yes: 6xYes, (x + 3)^2
x^2 + 13x + 36Yes: x and 6No: 12x is neededNo
x^2 + 6x - 9No: the last term is negative—No
9x^2 - 30xy + 25y^2Yes: 3x and 5yYes: 30xyYes, (3x - 5y)^2

If a trinomial is not a perfect square, use the methods in Factoring Trinomials or Factoring Trinomials When a Is Not 1.

Finding the missing last term

Sometimes you need to make a perfect square. For x^2 + 8x + c, take half of 8 and square it: 4^2 = 16. So x^2 + 8x + 16 = (x + 4)^2. This idea is the start of completing the square, which comes in the quadratics unit.

Practice Problems

  1. Factor x^2 + 6x + 9.
  2. Factor x^2 - 16x + 64.
  3. Factor m^2 + 2m + 1.
  4. Factor 25x^2 + 20x + 4.
  5. Factor 16x^2 - 24x + 9.
  6. Factor a^2 + 10ab + 25b^2.
  7. Is x^2 + 10x + 16 a perfect square trinomial? Factor it.
  8. Factor 3x^2 + 30x + 75 completely.
  9. Find c so that x^2 - 18x + c is a perfect square trinomial. Then factor it.
  10. A square photo has an area of 4x^2 + 28x + 49 square inches. Find the length of one side. Then find the side and the area when x = 3.
Show answers
  1. (x + 3)^2
  2. (x - 8)^2
  3. (m + 1)^2
  4. (5x + 2)^2. Check: 2 \cdot 5x \cdot 2 = 20x.
  5. (4x - 3)^2. Check: 2 \cdot 4x \cdot 3 = 24x.
  6. (a + 5b)^2
  7. No. The ends are squares of x and 4, but 2 \cdot x \cdot 4 = 8x, not 10x. It factors as (x + 2)(x + 8).
  8. 3(x^2 + 10x + 25) = 3(x + 5)^2
  9. Half of 18 is 9, and 9^2 = 81. So c = 81, and x^2 - 18x + 81 = (x - 9)^2.
  10. (2x + 7)^2, so each side is 2x + 7 inches. When x = 3: the side is 13 inches and the area is 169 square inches.

FAQ

How do you know if a trinomial is a perfect square?

The first and last terms must be perfect squares, and the last term must be positive. Then the middle term must equal 2ab, where a and b are the square roots of the ends.

What is the perfect square trinomial formula?

a^2 + 2ab + b^2 = (a + b)^2 and a^2 - 2ab + b^2 = (a - b)^2. The sign in the binomial matches the sign of the middle term.

Is a perfect square trinomial the same as a difference of squares?

No. A perfect square trinomial has three terms and factors into two equal binomials. A difference of squares has two terms and factors into a sum times a difference.

Can a perfect square trinomial have a negative last term?

No. The last term is b^2, and a square is never negative. So a trinomial like x^2 - 4x - 4 is not a perfect square.

Summary

  • A perfect square trinomial equals a binomial squared, like x^2 + 10x + 25 = (x + 5)^2.
  • Check that the first and last terms are squares and the last term is positive.
  • Check that the middle term is 2ab.
  • Use the sign of the middle term as the sign in the binomial.
  • Take out the GCF first. If the middle term doesn’t match, factor another way.

Previous lesson: Factoring the Difference of Squares

Next lesson: Factoring Completely

Back to the Algebra 1 Index

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