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Perfect Square Trinomials

Grade 9 · Algebra · Worksheet 1

  1. Mason is building a square-shaped platform for a stage production. The area of the platform (in square meters) is given by the expression 49x² + 28x + 4. Recognizing this as a perfect square trinomial, Mason wants to factor it to find the binomial expression representing the side length of the platform. What is the factored form of the expression? Answer: ______________
  2. Factor: 9x² - 30x + 25 = ? Answer: ______________
  3. Factor: 49x² + 112x + 64 = ? Answer: ______________
  4. Factor: 16x² - 40x + 25 = ? Answer: ______________
  5. A square garden is designed with side length (x + 7) meters. The area of the garden can be expressed as a perfect square trinomial. What is the expanded form of the area expression? Answer: ______________
  6. A physics student is modeling the trajectory of a projectile launched from a catapult. The height of the projectile above ground (in meters) at time t seconds is given by the equation h(t) = 9t² - 30t + 25. The student recognizes this as a perfect square trinomial and wants to factor it to determine the time when the projectile reaches its minimum height. What is the factored form of this quadratic expression? Answer: ______________
  7. 9x² - 30x + 25 = ? Answer: ______________
  8. Factor: 121x² + 154x + 49 = ? Answer: ______________
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Answer Key & Explanations

Perfect Square Trinomials · Grade 9 · Worksheet 1

  1. Mason is building a square-shaped platform for a stage production. The area of the platform (in square meters) is given by the expression 49x² + 28x + 4. Recognizing this as a perfect square trinomial, Mason wants to factor it to find the binomial expression representing the side length of the platform. What is the factored form of the expression? Answer: (7x + 2)^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(49x²) = 7x. Find the square root of the last term: sqrt(4) = 2.
    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(49x²) = 7x. Step 3: Find the square root of the last term: sqrt(4) = 2. Step 4: Check if the middle term matches 2ab: 2 × (7x) × (2) = 28x. This matches the given middle term. Step 5: Since all conditions are satisfied, the factored form is (7x + 2)². Step 6: Verify by expanding: (7x + 2)² = (7x)² + 2(7x)(2) + 2² = 49x² + 28x + 4. The factored form is (7x + 2)².

  2. Factor: 9x² - 30x + 25 = ? Answer: (3x - 5)² Solution: Check if the trinomial fits the perfect square pattern a² - 2ab + b² 9x² = (3x)², so a = 3x 25 = 5², so b = 5 Check middle term: 2 × (3x) × (5) = 30x, which matches -30x Since it fits the pattern a² - 2ab + b² = (a - b)² Therefore, 9x² - 30x + 25 = (3x - 5)² The answer is (3x - 5)².
    Full step-by-step solution

    Step 1: Check if the trinomial fits the perfect square pattern a² - 2ab + b² Step 2: 9x² = (3x)², so a = 3x Step 3: 25 = 5², so b = 5 Step 4: Check middle term: 2 × (3x) × (5) = 30x, which matches -30x Step 5: Since it fits the pattern a² - 2ab + b² = (a - b)² Step 6: Therefore, 9x² - 30x + 25 = (3x - 5)² The answer is (3x - 5)².

  3. 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.

  4. Factor: 16x² - 40x + 25 = ? Answer: (4x - 5)² Solution: Step 1: Identify the perfect squares: 16x² = (4x)² and 25 = 5² Step 2: Check the middle term: 2 × (4x) × 5 = 40x Step 3: Since the middle term is negative, use the pattern (a - b)² = a² - 2ab + b² Step 4: Substitute a = 4x and b = 5: (4x - 5)² Step 5: Verify by expanding: (4x - 5)² = 16x² - 40x…
    Full step-by-step solution

    Step 1: Identify the perfect squares: 16x² = (4x)² and 25 = 5² Step 2: Check the middle term: 2 × (4x) × 5 = 40x Step 3: Since the middle term is negative, use the pattern (a - b)² = a² - 2ab + b² Step 4: Substitute a = 4x and b = 5: (4x - 5)² Step 5: Verify by expanding: (4x - 5)² = 16x² - 40x + 25 The answer is (4x - 5)².

  5. A square garden is designed with side length (x + 7) meters. The area of the garden can be expressed as a perfect square trinomial. What is the expanded form of the area expression? Answer: x^2 + 14x + 49 Solution: The area of a square is calculated by squaring its side length. The side length is given as (x + 7) meters.
    Full step-by-step solution

    Step 1: The area of a square is calculated by squaring its side length. Step 2: The side length is given as (x + 7) meters. Step 3: Area = (x + 7)^2 Step 4: Expand using the formula (a + b)^2 = a^2 + 2ab + b^2 Step 5: Here, a = x and b = 7 Step 6: Area = x^2 + 2(x)(7) + 7^2 Step 7: Area = x^2 + 14x + 49 Step 8: The expanded form of the area expression is x^2 + 14x + 49

  6. A physics student is modeling the trajectory of a projectile launched from a catapult. The height of the projectile above ground (in meters) at time t seconds is given by the equation h(t) = 9t² - 30t + 25. The student recognizes this as a perfect square trinomial and wants to factor it to determine the time when the projectile reaches its minimum height. What is the factored form of this quadratic expression? Answer: (3t - 5)² Solution: Perfect square trinomials follow the pattern a² ± 2ab + b² = (a ± b)². This factoring method is useful in physics for analyzing projectile motion and finding optimal values.
    Full step-by-step solution

    Perfect square trinomials follow the pattern a² ± 2ab + b² = (a ± b)². To factor them, first verify that both the first and last terms are perfect squares, then check if the middle term equals ±2 times the product of their square roots. This factoring method is useful in physics for analyzing projectile motion and finding optimal values.

  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.

  8. Factor: 121x² + 154x + 49 = ? Answer: (11x + 7)² Solution: Identify the first term: 121x² = (11x)², so a = 11x. Identify the last term: 49 = 7², so b = 7. Check the middle term: 2 × (11x) × 7 = 154x, which matches the given +154x.
    Full step-by-step solution

    Step 1: Identify the first term: 121x² = (11x)², so a = 11x. Step 2: Identify the last term: 49 = 7², so b = 7. Step 3: Check the middle term: 2 × (11x) × 7 = 154x, which matches the given +154x. Step 4: Since the middle term is positive, the trinomial fits the pattern a² + 2ab + b² = (a + b)². Step 5: Therefore, 121x² + 154x + 49 = (11x + 7)². Step 6: Verify by expanding: (11x + 7)² = (11x)² + 2(11x)(7) + 7² = 121x² + 154x + 49. The answer is (11x + 7)².