Perfect Square Trinomials

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Perfect Square Trinomials · Grade 9 · Worksheet 2

1. 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 9x² + 30x + 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: (3x + 5) Solution: Identify the perfect square trinomial pattern: a² + 2ab + b² = (a + b)² Find the square root of the first term: sqrt(9x²) = 3x Find the square root of the last term: sqrt(25) = 5 Check if the middle term equals 2 × (3x) × (5): 2 × 3x × 5 = 30x Since 30x matches the middle term, this is a perfect…
   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(9x²) = 3x Step 3: Find the square root of the last term: sqrt(25) = 5 Step 4: Check if the middle term equals 2 × (3x) × (5): 2 × 3x × 5 = 30x Step 5: Since 30x matches the middle term, this is a perfect square trinomial Step 6: Factor as (3x + 5)² Step 7: Since this represents the area of a square, the side length is (3x + 5) The exact dimensions of the solar panel are (3x + 5) centimeters.

2. Factor: x² - 14x + 49 = ? Answer: (x - 7)² Solution: Identify the pattern: x² - 14x + 49 Check if first term is perfect square: x² = (x)² Check if last term is perfect square: 49 = 7² Check if middle term equals 2 × first term root × last term root: 2 × x × 7 = 14x Since the middle term is negative, use the pattern (a - b)² = a² - 2ab + b²…
   Full step-by-step solution

   Step 1: Identify the pattern: x² - 14x + 49 Step 2: Check if first term is perfect square: x² = (x)² Step 3: Check if last term is perfect square: 49 = 7² Step 4: Check if middle term equals 2 × first term root × last term root: 2 × x × 7 = 14x Step 5: Since the middle term is negative, use the pattern (a - b)² = a² - 2ab + b² Step 6: Therefore, x² - 14x + 49 = (x - 7)²

3. Isabella is a landscape architect designing a square-shaped community garden. The area of the garden is given by the perfect square trinomial 49x² + 84x + 36 square meters, where x represents a variable length in meters related to the garden's design. Isabella needs to factor this expression to find the binomial that represents the side length of the square garden. What is the factored form of 49x² + 84x + 36? Answer: (7x + 6)^2 Solution: Recognize 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: Recognize 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 the middle term: 2 × (7x) × (6) = 84x, which matches the given middle term. Step 5: Since all conditions are satisfied, 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 factored form is (7x + 6)^2.

4. A landscape architect is designing a rectangular garden for a community park. The area of the garden can be modeled by the quadratic expression 4x² + 20x + 25 square meters. She needs to determine the side lengths of the garden to order materials. If the garden's length and width are binomial expressions in terms of x, what is the factored form of the area expression that reveals these dimensions? Answer: (2x + 5)^2 Solution: Area = 4x² + 20x + 25 where a = 4, b = 20, c = 25. ( m x + n )² = m² x² + 2 m n x + n².
   Full step-by-step solution

   Let's factor the quadratic expression step by step. We are given: Area = 4x² + 20x + 25 --- **Step 1: Recognize the form** The expression is of the form: a x² + b x + c where a = 4, b = 20, c = 25. --- **Step 2: Check if it’s a perfect square trinomial** A perfect square trinomial has the form: ( m x + n )² = m² x² + 2 m n x + n². Compare with 4x² + 20x + 25: - m² = 4 → m = 2 (taking positive for dimensions) - n² = 25 → n = 5 - Check middle term: 2 m n = 2 * 2 * 5 = 20, which matches b = 20. --- **Step 3: Write the factored form** Since m = 2, n = 5, and 2 m n = 20 matches, we have: 4x² + 20x + 25 = (2x + 5)². --- **Step 4: Interpret the result** The factored form (2x + 5)² means the garden is a square with side length (2x + 5) meters. --- **Final answer:** (2x + 5)^2

5. 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.

6. A rectangular garden has an area that can be expressed as 4x² + 20x + 25 square meters. The landscape architect needs to determine the side lengths for ordering fencing materials. If the garden's length and width are binomial expressions with integer coefficients, what is the factored form that represents the side lengths? Answer: (2x + 5)^2 Solution: Area = 4x² + 20x + 25 a = 4, b = 20, c = 25 We want to factor it into two binomials (length and width) with integer coefficients.
   Full step-by-step solution

   Let's go step by step. We are given the area: Area = 4x² + 20x + 25 --- **Step 1: Recognize the form** The expression is a quadratic trinomial: a = 4, b = 20, c = 25 We want to factor it into two binomials (length and width) with integer coefficients. --- **Step 2: Check for perfect square trinomial** A perfect square trinomial has the form: (px + q)² = p²x² + 2pq x + q² Compare with 4x² + 20x + 25: - p² = 4 → p = 2 (taking positive integer for simplicity) - q² = 25 → q = 5 (taking positive integer) - Check middle term: 2pq = 2 × 2 × 5 = 20, which matches the given middle term 20x. --- **Step 3: Write the factorization** Since it matches perfectly: 4x² + 20x + 25 = (2x + 5)² --- **Step 4: Interpret the result** (2x + 5)² means both the length and width are (2x + 5) meters — the garden is a square. --- **Final Answer:** (2x + 5)^2

7. 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).