Equivalent Forms

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Equivalent Forms · Grade 9 · Worksheet 3

1. Liam is designing a rectangular garden with a perimeter of 60 meters. He wants to model the area of the garden as a function of its width. If the width is represented by w meters, what is the area function A(w) in standard quadratic form? Answer: A(w) = -w^2 + 30w Solution: We have a rectangular garden with perimeter \( P = 60 \) meters. Let \( w \) = width in meters, and \( l \) = length in meters.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangular garden with perimeter \( P = 60 \) meters. Let \( w \) = width in meters, and \( l \) = length in meters. The perimeter formula for a rectangle is: \[ P = 2w + 2l \] Substitute \( P = 60 \): \[ 2w + 2l = 60 \] --- **Step 2: Solve for length in terms of width** Divide both sides by 2: \[ w + l = 30 \] So: \[ l = 30 - w \] --- **Step 3: Write the area formula** Area \( A \) of a rectangle is: \[ A = w \times l \] Substitute \( l = 30 - w \): \[ A(w) = w \times (30 - w) \] --- **Step 4: Expand to standard quadratic form** \[ A(w) = 30w - w^2 \] Rewriting in standard form \( ax^2 + bx + c \): \[ A(w) = -w^2 + 30w \] --- **Step 5: Final answer** \[ A(w) = -w^2 + 30w \] This is the area as a function of width \( w \) in standard quadratic form.

2. (3x² - 5x + 2) + (2x² + 7x - 4) = ? Answer: 5x² + 2x - 2 Solution: (3x² - 5x + 2) + (2x² + 7x - 4) Remove parentheses (since it's addition, the signs stay the same). 3x² - 5x + 2 + 2x² + 7x - 4 Group like terms together.
   Full step-by-step solution

   Let's add the two polynomials step by step. We have: (3x² - 5x + 2) + (2x² + 7x - 4) Step 1: Remove parentheses (since it's addition, the signs stay the same). 3x² - 5x + 2 + 2x² + 7x - 4 Step 2: Group like terms together. x² terms: 3x² + 2x² x terms: -5x + 7x constant terms: 2 - 4 Step 3: Combine the x² terms. 3x² + 2x² = (3 + 2)x² = 5x² Step 4: Combine the x terms. -5x + 7x = (-5 + 7)x = 2x Step 5: Combine the constants. 2 - 4 = -2 Step 6: Write the final polynomial. 5x² + 2x - 2 That is the simplified result.

3. Which form of the quadratic function y = x² - 14x + 49 reveals that it has a double root at x = 7? Answer: y = (x - 7)² Solution: Start with the standard form y = x² - 14x + 49. Recognize that this is a perfect square trinomial because the first term is x², the last term is 49 = 7², and the middle term is -14x = 2(x)(-7).
   Full step-by-step solution

   Step 1: Start with the standard form y = x² - 14x + 49. Step 2: Recognize that this is a perfect square trinomial because the first term is x², the last term is 49 = 7², and the middle term is -14x = 2(x)(-7). Step 3: Factor the trinomial as (x - 7)(x - 7) = (x - 7)². Step 4: The factored form y = (x - 7)² clearly shows that the only zero is x = 7 (a double root), because setting (x - 7)² = 0 gives x - 7 = 0, so x = 7. The answer is y = (x - 7)².

4. 2^(x+1) - 2^(x-1) = 12 Answer: 3 Solution: Rewrite the equation: 2^(x+1) - 2^(x-1) = 12 Factor out 2^(x-1): 2^(x-1)(2^2 - 1) = 12 Simplify inside parentheses: 2^(x-1)(4 - 1) = 12 Calculate: 2^(x-1)(3) = 12 Divide both sides by 3: 2^(x-1) = 4 Rewrite 4 as a power of 2: 2^(x-1) = 2^2 Set exponents equal: x - 1 = 2 Solve for x: x = 3 The…
   Full step-by-step solution

   Step 1: Rewrite the equation: 2^(x+1) - 2^(x-1) = 12 Step 2: Factor out 2^(x-1): 2^(x-1)(2^2 - 1) = 12 Step 3: Simplify inside parentheses: 2^(x-1)(4 - 1) = 12 Step 4: Calculate: 2^(x-1)(3) = 12 Step 5: Divide both sides by 3: 2^(x-1) = 4 Step 6: Rewrite 4 as a power of 2: 2^(x-1) = 2^2 Step 7: Set exponents equal: x - 1 = 2 Step 8: Solve for x: x = 3 The answer is 3.

5. √(x² - 6x + 9) = ? when x = 7 Answer: 4 Solution: Substitute x = 7 into the expression: √(7² - 6×7 + 9) Calculate inside the square root: 7² = 49, 6×7 = 42, so 49 - 42 + 9 = 16 Simplify the square root: √16 = 4 The answer is 4.
   Full step-by-step solution

   Step 1: Substitute x = 7 into the expression: √(7² - 6×7 + 9) Step 2: Calculate inside the square root: 7² = 49, 6×7 = 42, so 49 - 42 + 9 = 16 Step 3: Simplify the square root: √16 = 4 The answer is 4.

6. Rewrite the expression 16x² - 40x + 25 in a form that shows it is a perfect square trinomial. Answer: (4x - 5)² Solution: Recognize the form of a perfect square trinomial: a² - 2ab + b² = (a - b)². The first term is 16x², which is (4x)², so a = 4x. The last term is 25, which is 5², so b = 5.
   Full step-by-step solution

   Step 1: Recognize the form of a perfect square trinomial: a² - 2ab + b² = (a - b)². Step 2: The first term is 16x², which is (4x)², so a = 4x. Step 3: The last term is 25, which is 5², so b = 5. Step 4: Check the middle term: -2ab = -2(4x)(5) = -40x, which matches the given middle term -40x. Step 5: Therefore, the expression is a perfect square trinomial: (4x - 5)². The answer is (4x - 5)².

7. Rewrite 4x² + 20x + 25 in a form that shows it is a perfect square trinomial. Answer: (2x + 5)² Solution: Recognize the form of a perfect square trinomial: (ax + b)² = a²x² + 2abx + b². Compare with 4x² + 20x + 25. The first term is 4x², so a² = 4, giving a = 2 (since a > 0).
   Full step-by-step solution

   Step 1: Recognize the form of a perfect square trinomial: (ax + b)² = a²x² + 2abx + b². Step 2: Compare with 4x² + 20x + 25. The first term is 4x², so a² = 4, giving a = 2 (since a > 0). Step 3: The last term is 25, so b² = 25, giving b = 5 (since b > 0). Step 4: Check the middle term: 2abx = 2(2)(5)x = 20x, which matches the given middle term. Step 5: Therefore, 4x² + 20x + 25 = (2x + 5)². The answer is (2x + 5)².

8. √(x² - 6x + 9) = ? when x = 8 Answer: 5 Solution: Identify the expression under the square root: x² - 6x + 9. Recognize that this is a perfect square trinomial: (x - 3)². Rewrite the problem: √((x - 3)²).
   Full step-by-step solution

   Step 1: Identify the expression under the square root: x² - 6x + 9. Step 2: Recognize that this is a perfect square trinomial: (x - 3)². Step 3: Rewrite the problem: √((x - 3)²). Step 4: The square root of a square gives the absolute value: |x - 3|. Step 5: Substitute x = 8: |8 - 3| = |5| = 5. The answer is 5.

9. A city's population growth is modeled by the function P(t) = 8000(1.04)^t, where t is time in years. The city council wants to rewrite this in the form P(t) = 8000e^(kt) to compare with other cities' growth models. Which expression is equivalent to the original population function? Answer: C. 8000e^(t·ln(1.04)) Solution: The original function is P(t) = 8000(1.04)^t To rewrite in the form P(t) = 8000e^(kt), we need to express (1.04)^t as e^(kt) Using the property that a^t = e^(t·ln(a)) for any positive a Therefore, (1.04)^t = e^(t·ln(1.04)) So P(t) = 8000e^(t·ln(1.04)) Option A is incorrect because e^(1.04t)…
   Full step-by-step solution

   Step 1: The original function is P(t) = 8000(1.04)^t Step 2: To rewrite in the form P(t) = 8000e^(kt), we need to express (1.04)^t as e^(kt) Step 3: Using the property that a^t = e^(t·ln(a)) for any positive a Step 4: Therefore, (1.04)^t = e^(t·ln(1.04)) Step 5: So P(t) = 8000e^(t·ln(1.04)) Step 6: This matches option C Step 7: Option A is incorrect because e^(1.04t) would be equivalent to (e^1.04)^t, not (1.04)^t Step 8: Option B is incorrect because e^(0.04t) would be equivalent to (e^0.04)^t ≈ (1.0408)^t, not exactly (1.04)^t Step 9: Option D is incorrect because ln(0.04) is negative, which would represent population decay, not growth The correct answer is 8000e^(t·ln(1.04)).