Perfect Square Trinomials

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Answer Key & Explanations

Perfect Square Trinomials · Grade 9 · Worksheet 2

1. Liam is designing a rectangular garden with an area that can be expressed as the trinomial x² + 14x + 49 square feet. He realizes this area represents a perfect square. What is the factored form of this expression, which would give the side length of the square garden? Answer: (x+7)^2 Solution: Area = x² + 14x + 49 a² + 2ab + b² = (a + b)² a² - 2ab + b² = (a - b)² Here, the first term is x², so a = x. The middle term is 14x. In the formula a² + 2ab + b², the middle term is 2ab.
   Full step-by-step solution

   Let's factor the trinomial step by step. We are given: Area = x² + 14x + 49 --- **Step 1: Recognize the form of a perfect square trinomial** A perfect square trinomial has the form: a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)² Here, the first term is x², so a = x. --- **Step 2: Identify b** The middle term is 14x. In the formula a² + 2ab + b², the middle term is 2ab. So: 2 * a * b = 14x Substitute a = x: 2 * x * b = 14x Divide both sides by 2x: b = 7 --- **Step 3: Check the third term** The third term should be b² = 7² = 49. Our third term is 49, which matches. --- **Step 4: Write the factored form** Since a = x, b = 7, and the sign of the middle term is positive, we have: x² + 14x + 49 = (x + 7)² --- **Step 5: Conclusion** The factored form is (x + 7)², which means the side length of the square garden is (x + 7) feet. **Final answer:** (x+7)^2

2. A company's quarterly profit in thousands of dollars is modeled by the quadratic function P(x) = 4x² + 28x + 49, where x represents the quarter number. The company's financial analyst notices this can be factored as a perfect square trinomial. What is the value that completes the square: (2x + k)²? Answer: 7 Solution: A perfect square trinomial has the form (ax + b)² = a²x² + 2abx + b² Compare with P(x) = 4x² + 28x + 49 The coefficient of x² is 4, so a² = 4, which means a = 2 The constant term is 49, so b² = 49, which means b = 7 Check the middle term: 2ab = 2 × 2 × 7 = 28, which matches the given coefficient…
   Full step-by-step solution

   Step 1: A perfect square trinomial has the form (ax + b)² = a²x² + 2abx + b² Step 2: Compare with P(x) = 4x² + 28x + 49 Step 3: The coefficient of x² is 4, so a² = 4, which means a = 2 Step 4: The constant term is 49, so b² = 49, which means b = 7 Step 5: Check the middle term: 2ab = 2 × 2 × 7 = 28, which matches the given coefficient Step 6: Therefore, P(x) = (2x + 7)², so k = 7 The answer is 7.

3. Ava is a structural engineer designing a square-shaped steel support platform for a bridge. The area of the platform (in square meters) is given by the perfect square trinomial 144x² + 120x + 25, where x represents a scaling factor for the platform's dimensions. Ava needs to factor this expression to find the binomial that represents the side length of the square platform. What is the factored form of 144x² + 120x + 25? Answer: (12x + 5)^2 Solution: Identify the pattern of a perfect square trinomial: a² + 2ab + b² = (a + b)². Find the square root of the first term: sqrt(144x²) = 12x. Find the square root of the last term: sqrt(25) = 5.
   Full step-by-step solution

   Step 1: Identify the pattern of a perfect square trinomial: a² + 2ab + b² = (a + b)². Step 2: Find the square root of the first term: sqrt(144x²) = 12x. Step 3: Find the square root of the last term: sqrt(25) = 5. Step 4: Check if the middle term matches 2ab: 2 × (12x) × (5) = 120x. The given middle term is 120x, which matches 2ab. Step 5: Since all conditions are satisfied, the factored form is (12x + 5)². Step 6: Verify by expanding: (12x + 5)² = (12x)² + 2(12x)(5) + 5² = 144x² + 120x + 25. This matches the original trinomial. The factored form is (12x + 5)².

4. A square courtyard has an area represented by the polynomial expression 4x² + 20x + 25 square meters. The courtyard is being redesigned with a decorative border, and the architect needs to determine the side length expression to calculate materials needed. What binomial expression represents the length of one side of this square courtyard? Answer: (2x + 5) Solution: Recognize that the area of a square courtyard is given by 4x² + 20x + 25 Since it's a square, the area equals (side length)² Check if this is a perfect square trinomial by examining the first and last terms The square root of 4x² is 2x The square root of 25 is 5 Check if the middle term equals 2…
   Full step-by-step solution

   Step 1: Recognize that the area of a square courtyard is given by 4x² + 20x + 25 Step 2: Since it's a square, the area equals (side length)² Step 3: Check if this is a perfect square trinomial by examining the first and last terms Step 4: The square root of 4x² is 2x Step 5: The square root of 25 is 5 Step 6: Check if the middle term equals 2 × (2x) × (5) = 20x Step 7: Since 20x matches the middle term, this is indeed a perfect square trinomial Step 8: Therefore, the factored form is (2x + 5)² Step 9: The side length of the square courtyard is (2x + 5) meters

5. A square garden has an area represented by the expression 4x² + 20x + 25 square meters. If the garden's side length is expressed as (ax + b) meters where a and b are positive integers, determine the factored form that represents the side length of this square garden. Answer: (2x+5) Solution: Step 1: Identify the perfect square trinomial: 4x² + 20x + 25 Step 2: Check if first term is a perfect square: sqrt(4x²) = 2x Step 3: Check if last term is a perfect square: sqrt(25) = 5 Step 4: Verify the middle term: 2 × (2x) × (5) = 20x Step 5: Since the middle term matches, we can factor as:…
   Full step-by-step solution

   Step 1: Identify the perfect square trinomial: 4x² + 20x + 25 Step 2: Check if first term is a perfect square: sqrt(4x²) = 2x Step 3: Check if last term is a perfect square: sqrt(25) = 5 Step 4: Verify the middle term: 2 × (2x) × (5) = 20x Step 5: Since the middle term matches, we can factor as: (2x + 5)² Step 6: For a square garden, the side length is (2x + 5) meters The factored form representing the side length is (2x+5).

6. x² + 10x + 25 = ? Answer: (x+5)² Solution: 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.
   Full step-by-step solution

   Step 1: Identify the perfect square pattern: x² is (x)² and 25 is (5)² Step 2: Check if the middle term equals 2 × (first term root) × (last term root): 2 × x × 5 = 10x Step 3: Since all conditions are met, factor as (first term root + last term root)² Step 4: Write the factored form: (x + 5)² The answer is (x+5)².

7. Olivia is designing a square-shaped mural for a community center. The area of the mural (in square meters) is given by the trinomial 25x² + 60x + 36, where x represents the number of additional meters added to the original side length. Olivia recognizes this as a perfect square trinomial and needs to factor it to determine the side length of the square mural. What is the factored form of this expression? Answer: (5x + 6)^2 Solution: Identify the pattern of a perfect square trinomial: a² + 2ab + b² = (a + b)². Find the square root of the first term: sqrt(25x²) = 5x. Find the square root of the last term: sqrt(36) = 6.
   Full step-by-step solution

   Step 1: Identify the pattern of a perfect square trinomial: a² + 2ab + b² = (a + b)². Step 2: Find the square root of the first term: sqrt(25x²) = 5x. Step 3: Find the square root of the last term: sqrt(36) = 6. Step 4: Check if the middle term matches 2ab: 2 × (5x) × (6) = 60x. The middle term is 60x, which matches. Step 5: Since all conditions are satisfied, the factored form is (5x + 6)². Step 6: Verify by expanding: (5x + 6)² = (5x)² + 2(5x)(6) + 6² = 25x² + 60x + 36. This matches the original trinomial. The factored form is (5x + 6)².