Perfect Square Trinomials

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Perfect Square Trinomials · Grade 9 · Worksheet 3

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 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, what is the factored form that represents the side length? Answer: (2x+5) Solution: Identify the perfect square trinomial: 4x² + 20x + 25 Check if the first term is a perfect square: sqrt(4x²) = 2x Check if the last term is a perfect square: sqrt(25) = 5 Check if the middle term equals 2 × (2x) × (5) = 20x Since all conditions are met, factor as (2x + 5)² Therefore, the side…
   Full step-by-step solution

   Step 1: Identify the perfect square trinomial: 4x² + 20x + 25 Step 2: Check if the first term is a perfect square: sqrt(4x²) = 2x Step 3: Check if the last term is a perfect square: sqrt(25) = 5 Step 4: Check if the middle term equals 2 × (2x) × (5) = 20x Step 5: Since all conditions are met, factor as (2x + 5)² Step 6: Therefore, the side length is (2x + 5) meters The answer is (2x+5).

3. A square mosaic is being designed for a community center. The total number of small colored tiles needed to create the mosaic is given by the expression 4x² + 20x + 25, where x represents the number of tiles along one edge of a smaller decorative border. If the mosaic forms a perfect square arrangement, what is the factored form that represents the side length of the complete mosaic in terms of x? Answer: (2x+5)^2 Solution: Identify the perfect square trinomial: 4x² + 20x + 25 Recognize that the first term 4x² is (2x)² Recognize that the last term 25 is 5² Check if the middle term equals 2 × (first term root) × (last term root): 2 × (2x) × 5 = 20x Since all conditions are met, factor as (2x + 5)² Therefore, the…
   Full step-by-step solution

   Step 1: Identify the perfect square trinomial: 4x² + 20x + 25 Step 2: Recognize that the first term 4x² is (2x)² Step 3: Recognize that the last term 25 is 5² Step 4: Check if the middle term equals 2 × (first term root) × (last term root): 2 × (2x) × 5 = 20x Step 5: Since all conditions are met, factor as (2x + 5)² Step 6: Therefore, the side length of the mosaic is (2x + 5) tiles Step 7: The factored form representing the side length is (2x + 5)²

4. 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.

5. A rectangular garden has an area that can be expressed as x² + 14x + 49 square meters. If the length of the garden is represented by (x + k) meters, find the value of k and determine the width of the garden in terms of x. Answer: k = 7, width = (x + 7) meters Solution: Area of rectangle = x² + 14x + 49 square meters Length = (x + k) meters Width = unknown, in terms of x. Area = Length × Width x² + 14x + 49 = (x + k) × Width We recognize x² + 14x + 49 as a perfect square trinomial.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We are given: Area of rectangle = x² + 14x + 49 square meters Length = (x + k) meters Width = unknown, in terms of x. We know: Area = Length × Width So: x² + 14x + 49 = (x + k) × Width --- **Step 2: Factor the area expression** We recognize x² + 14x + 49 as a perfect square trinomial. Check: x² + 14x + 49 = x² + 2 × 7 × x + 7² That matches the pattern: a² + 2ab + b² = (a + b)² So: x² + 14x + 49 = (x + 7)² --- **Step 3: Relate to length and width** Area = (x + 7)² = (x + k) × Width If we choose k = 7, then length = (x + 7) meters. Then: (x + 7)² = (x + 7) × Width Divide both sides by (x + 7): Width = x + 7 --- **Step 4: Conclusion** k = 7 Width = (x + 7) meters --- **Final answer:** k = 7, width = (x + 7) meters

6. Matiu is an artist creating a square mosaic for a public art installation. The area of the mosaic (in square meters) is modeled by the perfect square trinomial 121x² - 44x + 4, where x represents a scaling factor for the tile pattern. Matiu needs to factor this expression to determine the binomial expression for the side length of the square mosaic. What is the factored form of 121x² - 44x + 4? Answer: (11x - 2)^2 Solution: Recognize the pattern of a perfect square trinomial: a² - 2ab + b² = (a - b)². Find the square root of the first term: sqrt(121x²) = 11x. Find the square root of the last term: sqrt(4) = 2.
   Full step-by-step solution

   Step 1: Recognize the pattern of a perfect square trinomial: a² - 2ab + b² = (a - b)². Step 2: Find the square root of the first term: sqrt(121x²) = 11x. Step 3: Find the square root of the last term: sqrt(4) = 2. Step 4: Check if the middle term matches -2ab: -2 × (11x) × (2) = -44x. The middle term is -44x, which matches. Step 5: Since the middle term is negative, the factored form uses subtraction: (11x - 2)². Step 6: Verify by expanding: (11x - 2)² = (11x)² - 2(11x)(2) + 2² = 121x² - 44x + 4. This matches the original trinomial. The factored form is (11x - 2)².

7. 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.