Perfect Square Trinomials

Grade 9 · algebra · 100 practice problems · read aloud

🔊 Listen to this explanation

Perfect Square Trinomials

🔍 What is it and Why is it Useful?

A Perfect Square Trinomial is a quadratic expression created by squaring a binomial. It follows the pattern: (a + b)² = a² + 2ab + b² or (a - b)² = a² - 2ab + b².

Recognizing this pattern is a superpower in algebra! It lets you factor trinomials instantly and solve quadratic equations much faster. It's the foundation for completing the square, a key technique you'll use throughout high school math.

📝 Step-by-Step Guide

To identify and factor a perfect square trinomial:

Check the Form: Is it a trinomial (three terms)?
Check the First and Last Terms: Are they perfect squares? (e.g., x², 25, 1, 9y²)
Check the Middle Term: Does it equal 2 × √(first term) × √(last term)?
Determine the Sign: Look at the sign of the middle term.
Write the Square: It factors as (√(first term) ± √(last term))².

✨ Visual Examples

Example 1: Factor x² + 6x + 9

First term: √(x²) = x
Last term: √9 = 3
Middle term check: 2 * (x) * (3) = 6x ✅
Sign is positive.
Answer: (x + 3)²

Example 2: Factor 4y² - 20y + 25

First term: √(4y²) = 2y
Last term: √25 = 5
Middle term check: 2 * (2y) * (5) = 20y ✅ (The sign is negative)
Sign is negative.
Answer: (2y - 5)²

⚠️ Common Mistakes

Forgetting the "2" in the middle term. Students often just add the square roots without checking if the middle term is truly 2ab. For x² + 5x + 25, the middle term is 5x, but 2*x*5 = 10x. Since 5x ≠ 10x, it's not a perfect square trinomial!

Mishandling the sign. The sign in the binomial always matches the sign of the middle term in the trinomial. If the middle term is negative, use a minus sign: a² - 2ab + b² = (a - b)².

💡 Tips & Tricks

The "I SEE A" Pattern: A great memory aid!
I - Is the first term a perfect square?
C - Is the last term a perfect square?
E - Is the middle term Equal to 2ab?
A - Answer is (a ± b)²!

Shortcut: Once you confirm it's a perfect square, just write the square root of the first term, the sign from the middle term, the square root of the last term, and square the whole thing.

🏋️ Practice Suggestions

Start Simple: Practice squaring binomials like (x+2)² to see the pattern in action.
Flashcards: Create flashcards with trinomials on one side and "Perfect Square?" on the back. Sort them into "Yes" and "No" piles.
Error Analysis: Find and correct the mistake in incorrect factorings, like "x² + 4x + 4 = (x + 2)(x + 2)" is correct, but "x² + 4x + 4 = (x + 4)²" is wrong. Why?

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 x² + 6x + 9 = ?

Hint: Look for a pattern where the first and last terms are perfect squares, and the middle term is twice the product of their square roots.

Show the answer

Answer: (x+3)²

Recognize that this is a quadratic trinomial of the form: a x² + b x + c Here, a = 1, b = 6, c = 9.
Check if it is a perfect square trinomial. A perfect square trinomial has the form: (x + m)² = x² + 2 m x + m². Compare coefficients: From x² + 6x + 9, the middle term coefficient is 6, so 2 m = 6. Thus m = 3.
Check the constant term: m² should equal 9. m = 3 gives m² = 9, which matches the constant term in the expression.
Write the factored form: Since x² + 6x + 9 = x² + 2 * 3 * x + 3², it matches (x + 3)².
Verify by expanding (x + 3)²: (x + 3)(x + 3) = x*x + 3*x + 3*x + 3*3 = x² + 3x + 3x + 9 = x² + 6x + 9. This matches the original expression. Final answer: (x+3)²

We start with the expression: x² + 6x + 9

2 x² - 14x + 49 = ?

Hint: Look for a pattern where the first and last terms are perfect squares, and the middle term is twice the product of their square roots.

Show the answer

Answer: (x - 7)²

Identify the pattern: x² - 14x + 49
Check if first term is perfect square: x² = (x)²
Check if last term is perfect square: 49 = 7²
Check middle term: 14x = 2 × x × 7
Since all conditions are met, this is a perfect square trinomial
Write as (x - 7)²

The answer is (x - 7)².

3 x² + 14x + 49 = ?

Hint: This expression follows the pattern of a perfect square trinomial. Look for two numbers that multiply to the constant term and add to the coefficient of the middle term.

Show the answer

Answer: (x+7)²

Identify the pattern a² + 2ab + b² = (a + b)²
x² is a², so a = x
49 is b², so b = 7
Check the middle term: 2ab = 2 × x × 7 = 14x, which matches
Therefore, x² + 14x + 49 = (x + 7)²

The answer is (x+7)².

4 x² + 10x + 25 = ?

Hint: Look for a pattern where the first and last terms are perfect squares, and the middle term is twice the product of their square roots.

Show the answer

Answer: (x+5)²

Identify the perfect square pattern: x² is (x)² and 25 is (5)²
Check if the middle term equals 2 × (first term root) × (last term root): 2 × x × 5 = 10x
Since all conditions are met, factor as (first term root + last term root)²
Write the factored form: (x + 5)²

The answer is (x+5)².

5 x² - 10x + 25 = ?

Hint: Look for a trinomial that can be written as a squared binomial. Check if the first and last terms are perfect squares.

Show the answer

Answer: (x - 5)²

Identify the trinomial: x² - 10x + 25
Check if the first term is a perfect square: x² = (x)²
Check if the last term is a perfect square: 25 = (5)²
Check if the middle term equals 2 × (first term root) × (last term root): 2 × x × 5 = 10x
Since the middle term is -10x, the binomial will be (x - 5)
Write as a perfect square: (x - 5)²

The answer is (x - 5)².

6 4x² - 20x + 25 = ?

Hint: Look for a trinomial where the first and last terms are perfect squares, and the middle term is twice the product of their square roots.

Show the answer

Answer: (2x - 5)²

Identify if the trinomial is a perfect square. The first term 4x² is (2x)², and the last term 25 is 5².
Check the middle term: 2 × (2x) × 5 = 20x, which matches the absolute value of -20x.
Since the middle term is negative, the factored form is (2x - 5)².
Verify by expanding: (2x - 5)(2x - 5) = 4x² - 10x - 10x + 25 = 4x² - 20x + 25.

The answer is (2x - 5)².

Practise this topic — 10 free problems, no signup →