Perfect Square Trinomials

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Answer Key & Explanations

Perfect Square Trinomials · Grade 9 · Worksheet 1

1. Factor: 25x² - 60x + 36 = ? Answer: (5x - 6)² Solution: Identify if this is a perfect square trinomial. The first term 25x² is (5x)². The last term 36 is 6².
   Full step-by-step solution

   Step 1: Identify if this is a perfect square trinomial. The first term 25x² is (5x)². The last term 36 is 6². Step 2: Check the middle term: 2 × (5x) × (6) = 2 × 5x × 6 = 60x. Step 3: Since the middle term is negative (-60x), the factored form is (5x - 6)². Step 4: Verify: (5x - 6)² = (5x - 6)(5x - 6) = 25x² - 30x - 30x + 36 = 25x² - 60x + 36. The answer is (5x - 6)².

2. 49x² + 112x + 64 = ? Answer: (7x + 8)² Solution: Identify the first term: 49x² = (7x)², so a = 7x. Identify the last term: 64 = 8², so b = 8. Check the middle term: 2 × (7x) × 8 = 112x, which matches the given +112x.
   Full step-by-step solution

   Step 1: Identify the first term: 49x² = (7x)², so a = 7x. Step 2: Identify the last term: 64 = 8², so b = 8. Step 3: Check the middle term: 2 × (7x) × 8 = 112x, which matches the given +112x. Step 4: Since the middle term is positive, the factored form is (7x + 8)². Step 5: Verify by expanding: (7x + 8)² = (7x)² + 2(7x)(8) + 8² = 49x² + 112x + 64. The answer is (7x + 8)².

3. An engineer is designing a square solar panel array for a new satellite. The total area of the array can be modeled by the expression 9x² + 42x + 49 square centimeters, where x represents the additional length needed for mounting hardware. Recognizing this as a perfect square trinomial, the engineer needs to factor it to determine the side length of the square array. What is the factored form of this expression? Answer: (3x+7)^2 Solution: Identify the perfect squares in the trinomial 9x² + 42x + 49 - The first term 9x² is (3x)² - The last term 49 is 7² Check if the middle term equals 2 × (first square root) × (last square root) - 2 × (3x) × 7 = 2 × 21x = 42x Write the factored form using the pattern a² + 2ab + b² = (a + b)² - a =…
   Full step-by-step solution

   Step 1: Identify the perfect squares in the trinomial 9x² + 42x + 49 - The first term 9x² is (3x)² - The last term 49 is 7² Step 2: Check if the middle term equals 2 × (first square root) × (last square root) - 2 × (3x) × 7 = 2 × 21x = 42x - This matches the middle term exactly Step 3: Write the factored form using the pattern a² + 2ab + b² = (a + b)² - a = 3x, b = 7 - Therefore, 9x² + 42x + 49 = (3x + 7)² The factored form is (3x + 7)².

4. 16x² - 40x + 25 = ? Answer: (4x - 5)² Solution: Identify if the trinomial is a perfect square. The first term is 16x², which is (4x)². The last term is 25, which is 5².
   Full step-by-step solution

   Step 1: Identify if the trinomial is a perfect square. The first term is 16x², which is (4x)². The last term is 25, which is 5². Step 2: Check the middle term: 2 × (4x) × (5) = 2 × 20x = 40x. Since our middle term is -40x, we have the negative form. Step 3: Write as a squared binomial: (4x - 5)² Step 4: Verify by expanding: (4x - 5)² = (4x - 5)(4x - 5) = 16x² - 20x - 20x + 25 = 16x² - 40x + 25 Step 5: The factorization is correct. The answer is (4x - 5)².

5. A robotics team is designing a square-shaped solar panel for their competition robot. The area of the panel can be expressed as the trinomial 4x² + 20x + 25 square centimeters. If the side length of the square panel is represented by a binomial expression (ax + b), find the exact dimensions of the solar panel. Answer: (2x + 5) centimeters Solution: We are told the solar panel is square-shaped, so its area equals (side length)². The area is given as: 4x² + 20x + 25. The side length is a binomial (ax + b).
   Full step-by-step solution

   Step 1: Understand the problem We are told the solar panel is square-shaped, so its area equals (side length)². The area is given as: 4x² + 20x + 25. The side length is a binomial (ax + b). So: (ax + b)² = 4x² + 20x + 25. --- Step 2: Expand the square of the binomial (ax + b)² = a²x² + 2ab x + b². --- Step 3: Match coefficients with the given trinomial We have: a²x² + 2ab x + b² = 4x² + 20x + 25. Matching coefficients: - Coefficient of x²: a² = 4 → a = 2 or a = -2. Since side length is a physical dimension, we take a = 2 (positive). - Constant term: b² = 25 → b = 5 or b = -5. Again, positive length → b = 5. - Coefficient of x: 2ab = 20. Check: 2 × 2 × 5 = 20. ✓ --- Step 4: Write the binomial ax + b = 2x + 5. --- Step 5: Conclusion The side length of the solar panel is (2x + 5) centimeters.

6. Noah is designing a square-shaped mosaic for an art installation. The area of the mosaic is given by the trinomial 49x² + 84x + 36 square inches, where x represents the number of tiles added to the original side length. Recognizing this as a perfect square trinomial, Noah wants to factor it to find the binomial expression that represents the side length of the square. What is the factored form of 49x² + 84x + 36? Answer: (7x + 6)^2 Solution: Identify the perfect square trinomial pattern: a² + 2ab + b² = (a + b)². Find the square root of the first term: sqrt(49x²) = 7x. Find the square root of the last term: sqrt(36) = 6.
   Full step-by-step solution

   Step 1: Identify the perfect square trinomial pattern: a² + 2ab + b² = (a + b)². Step 2: Find the square root of the first term: sqrt(49x²) = 7x. Step 3: Find the square root of the last term: sqrt(36) = 6. Step 4: Check if the middle term matches 2ab: 2 × (7x) × (6) = 84x. Since the middle term is 84x, the condition is satisfied. Step 5: Since both the first and last terms are perfect squares and the middle term is twice the product of their square roots, the expression is a perfect square trinomial. The factored form is (7x + 6)². Step 6: Verify by expanding: (7x + 6)² = (7x)² + 2(7x)(6) + 6² = 49x² + 84x + 36. The answer is (7x + 6)^2.

7. Mason is designing a square patio. The area of the patio is given by the expression 9x² + 42x + 49 square feet. If the side length is written as (ax + b) feet with a and b positive integers, what binomial expression represents the side length of the square patio? Answer: (3x + 7) Solution: Recognize that the area of a square patio is side length squared, so we need to factor 9x² + 42x + 49 as (ax + b)². A perfect square trinomial follows the pattern (ax + b)² = a²x² + 2abx + b².
   Full step-by-step solution

   Step 1: Recognize that the area of a square patio is side length squared, so we need to factor 9x² + 42x + 49 as (ax + b)². Step 2: A perfect square trinomial follows the pattern (ax + b)² = a²x² + 2abx + b². Step 3: Compare with the given expression 9x² + 42x + 49. Step 4: Find a² = 9, so a = 3 (since a is positive). Step 5: Find b² = 49, so b = 7 (since b is positive). Step 6: Check the middle term: 2ab = 2 × 3 × 7 = 42, which matches the middle term 42x. Step 7: Therefore, the factored form is (3x + 7)². Step 8: The side length of the square patio is (3x + 7) feet. The answer is (3x + 7).