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Perfect Square Trinomials

Grade 9 · Algebra · Worksheet 3

  1. Charlotte is designing a square-shaped community garden. The city planning department requires that the total area of the garden (in square meters) be expressed as the trinomial 81x² + 144x + 64, where x represents the number of additional meters added to the original side length. Charlotte recognizes this as a perfect square trinomial and needs to factor it to determine the side length of the square garden. What is the factored form of this expression? Answer: ______________
  2. Kaia is a structural engineer designing a square-shaped support platform for a large sculpture. The area of the platform (in square meters) is given by the perfect square trinomial 144x² - 264x + 121, where x represents a scaling factor for the dimensions. Kaia 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 - 264x + 121? Answer: ______________
  3. A square-shaped mosaic floor in a museum lobby has an area represented by the expression 9x² + 42x + 49 square feet. The mosaic is made up of smaller square tiles arranged in a grid pattern. If the side length of the entire mosaic can be written as (ax + b) feet, where a and b are positive odd integers, what is the binomial expression for the side length? Answer: ______________
  4. Noah is designing a square-shaped reflecting pool for a new city park. The area of the pool (in square feet) is given by the perfect square trinomial 100x² - 180x + 81, where x represents a scaling factor in feet related to the pool's dimensions. Noah needs to factor this expression to find the binomial that represents the side length of the square pool. What is the factored form of 100x² - 180x + 81? Answer: ______________
  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: ______________
  6. Factor: 49x² + 112x + 64 = ? Answer: ______________
  7. 9x² - 30x + 25 = ? Answer: ______________
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Answer Key & Explanations

Perfect Square Trinomials · Grade 9 · Worksheet 3

  1. Charlotte is designing a square-shaped community garden. The city planning department requires that the total area of the garden (in square meters) be expressed as the trinomial 81x² + 144x + 64, where x represents the number of additional meters added to the original side length. Charlotte recognizes this as a perfect square trinomial and needs to factor it to determine the side length of the square garden. What is the factored form of this expression? Answer: (9x + 8)^2 Solution: Identify the perfect square trinomial pattern: a² + 2ab + b² = (a + b)². Find the square root of the first term: sqrt(81x²) = 9x. Find the square root of the last term: sqrt(64) = 8.
    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(81x²) = 9x. Step 3: Find the square root of the last term: sqrt(64) = 8. Step 4: Check if the middle term matches 2ab: 2 × (9x) × (8) = 144x. The middle term is 144x, which matches. Step 5: Since all conditions are satisfied, the factored form is (9x + 8)². Step 6: Verify by expanding: (9x + 8)² = (9x)² + 2(9x)(8) + 8² = 81x² + 144x + 64. This matches the original trinomial. The factored form is (9x + 8)².

  2. Kaia is a structural engineer designing a square-shaped support platform for a large sculpture. The area of the platform (in square meters) is given by the perfect square trinomial 144x² - 264x + 121, where x represents a scaling factor for the dimensions. Kaia 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 - 264x + 121? Answer: (12x - 11)^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(121) = 11.
    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(121) = 11. Step 4: Check if the middle term matches -2ab: 2 × (12x) × (11) = 264x. The given middle term is -264x, which matches -2ab. Step 5: Since all conditions are satisfied, the factored form is (12x - 11)². Step 6: Verify by expanding: (12x - 11)² = (12x)² - 2(12x)(11) + 11² = 144x² - 264x + 121. This matches the original trinomial. The factored form is (12x - 11)².

  3. A square-shaped mosaic floor in a museum lobby has an area represented by the expression 9x² + 42x + 49 square feet. The mosaic is made up of smaller square tiles arranged in a grid pattern. If the side length of the entire mosaic can be written as (ax + b) feet, where a and b are positive odd integers, what is the binomial expression for the side length? Answer: (3x + 7) Solution: Recognize that the area of a square is (side length)², so we need to factor 9x² + 42x + 49 as a perfect square trinomial. A perfect square trinomial has the form a²x² + 2abx + b² = (ax + b)².
    Full step-by-step solution

    Step 1: Recognize that the area of a square is (side length)², so we need to factor 9x² + 42x + 49 as a perfect square trinomial. Step 2: A perfect square trinomial has the form a²x² + 2abx + b² = (ax + b)². Step 3: Find the square root of the first term: sqrt(9x²) = 3x. Step 4: Find the square root of the last term: sqrt(49) = 7. Step 5: Check the middle term: 2ab = 2 × 3 × 7 = 42, which matches the given middle term 42x. Step 6: Since all conditions are satisfied, the trinomial is a perfect square. Step 7: Write the factored form: (3x + 7)². Step 8: The side length of the mosaic is 3x + 7 feet. The answer is (3x + 7).

  4. Noah is designing a square-shaped reflecting pool for a new city park. The area of the pool (in square feet) is given by the perfect square trinomial 100x² - 180x + 81, where x represents a scaling factor in feet related to the pool's dimensions. Noah needs to factor this expression to find the binomial that represents the side length of the square pool. What is the factored form of 100x² - 180x + 81? Answer: (10x - 9)^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(100x²) = 10x. Find the square root of the last term: sqrt(81) = 9.
    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(100x²) = 10x. Step 3: Find the square root of the last term: sqrt(81) = 9. Step 4: Check if the middle term matches -2ab: 2 × (10x) × (9) = 180x. The given middle term is -180x, which matches -2ab. Step 5: Since all conditions are satisfied, the factored form is (10x - 9)². Step 6: Verify by expanding: (10x - 9)² = (10x)² - 2(10x)(9) + 9² = 100x² - 180x + 81. This matches the original trinomial. The factored form is (10x - 9)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. Factor: 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 (a + b)² = (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.

  7. 9x² - 30x + 25 = ? Answer: (3x - 5)² Solution: Identify the first term: 9x² = (3x)² Identify the last term: 25 = 5² Check the middle term: -30x = 2 × (3x) × (-5) Since all conditions are met, this is a perfect square trinomial of the form (a - b)² Write the factored form: (3x - 5)² The answer is (3x - 5)².
    Full step-by-step solution

    Step 1: Identify the first term: 9x² = (3x)² Step 2: Identify the last term: 25 = 5² Step 3: Check the middle term: -30x = 2 × (3x) × (-5) Step 4: Since all conditions are met, this is a perfect square trinomial of the form (a - b)² Step 5: Write the factored form: (3x - 5)² The answer is (3x - 5)².