Completing the Square

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Completing the Square · Grade 10 · Worksheet 3

1. Complete the square: 3x² - 30x + 72 = 0 → (x + a)² = b, find a and b? Answer: a = -5, b = 1 Solution: Start with 3x² - 30x + 72 = 0 Divide both sides by 3: x² - 10x + 24 = 0 Move the constant term to the right side: x² - 10x = -24 Take half of the coefficient of x (which is -10): -10/2 = -5 Square it: (-5)² = 25 Add 25 to both sides: x² - 10x + 25 = -24 + 25 Simplify the right side: x² - 10x +…
   Full step-by-step solution

   Step 1: Start with 3x² - 30x + 72 = 0 Step 2: Divide both sides by 3: x² - 10x + 24 = 0 Step 3: Move the constant term to the right side: x² - 10x = -24 Step 4: Take half of the coefficient of x (which is -10): -10/2 = -5 Step 5: Square it: (-5)² = 25 Step 6: Add 25 to both sides: x² - 10x + 25 = -24 + 25 Step 7: Simplify the right side: x² - 10x + 25 = 1 Step 8: Factor the left side as a perfect square: (x - 5)² = 1 Step 9: Write in the form (x + a)² = b: (x + (-5))² = 1 Therefore, a = -5 and b = 1.

2. Complete the square: 2x² - 16x + 10 = 0 → (x + a)² = b, find a and b? Answer: a = -4, b = 11 Solution: Start with 2x² - 16x + 10 = 0 Factor out 2 from the x² and x terms: 2(x² - 8x) + 10 = 0 Move the constant term to the other side: 2(x² - 8x) = -10 Take half of the coefficient of x inside the parentheses: -8/2 = -4 Square it: (-4)² = 16 Add 16 inside the parentheses, but since it is multiplied…
   Full step-by-step solution

   Step 1: Start with 2x² - 16x + 10 = 0 Step 2: Factor out 2 from the x² and x terms: 2(x² - 8x) + 10 = 0 Step 3: Move the constant term to the other side: 2(x² - 8x) = -10 Step 4: Take half of the coefficient of x inside the parentheses: -8/2 = -4 Step 5: Square it: (-4)² = 16 Step 6: Add 16 inside the parentheses, but since it is multiplied by 2, add 2*16 = 32 to the right side: 2(x² - 8x + 16) = -10 + 32 Step 7: Simplify: 2(x - 4)² = 22 Step 8: Divide both sides by 2: (x - 4)² = 11 Step 9: This is in the form (x + a)² = b, so a = -4 and b = 11.

3. Complete the square: 2x² + 12x + 10 = 0 → (x + h)² = k, find h and k? Answer: h = 3, k = 4 Solution: Start with 2x² + 12x + 10 = 0 Divide both sides by 2 to simplify: x² + 6x + 5 = 0 Move the constant term to the right side: x² + 6x = -5 Take half of the coefficient of x (which is 6): 6/2 = 3 Square it: 3² = 9 Add 9 to both sides: x² + 6x + 9 = -5 + 9 Simplify the right side: x² + 6x + 9 = 4…
   Full step-by-step solution

   Step 1: Start with 2x² + 12x + 10 = 0 Step 2: Divide both sides by 2 to simplify: x² + 6x + 5 = 0 Step 3: Move the constant term to the right side: x² + 6x = -5 Step 4: Take half of the coefficient of x (which is 6): 6/2 = 3 Step 5: Square it: 3² = 9 Step 6: Add 9 to both sides: x² + 6x + 9 = -5 + 9 Step 7: Simplify the right side: x² + 6x + 9 = 4 Step 8: Factor the left side as a perfect square: (x + 3)² = 4 Step 9: Therefore, h = 3 and k = 4.

4. Mere is designing a parabolic mirror for a solar cooker. The cross-section of the mirror can be modeled by the quadratic function h(x) = -4x^2 + 24x - 20, where h(x) is the depth of the mirror in centimeters at a horizontal distance x centimeters from the left edge. Using the method of completing the square, rewrite the function in vertex form h(x) = a(x - p)^2 + q to determine the maximum depth of the mirror and the horizontal distance from the left edge where this maximum depth occurs. Then, state the coordinates of the vertex. Answer: Maximum depth of 16 cm at a horizontal distance of 3 cm; vertex at (3, 16) Solution: Start with the function: h(x) = -4x^2 + 24x - 20 Factor out -4 from the x^2 and x terms: h(x) = -4(x^2 - 6x) - 20 Complete the square inside the parentheses. Take half of -6, which is -3, and square it to get 9.
   Full step-by-step solution

   Step 1: Start with the function: h(x) = -4x^2 + 24x - 20 Step 2: Factor out -4 from the x^2 and x terms: h(x) = -4(x^2 - 6x) - 20 Step 3: Complete the square inside the parentheses. Take half of -6, which is -3, and square it to get 9. Step 4: Add and subtract 9 inside the parentheses: h(x) = -4(x^2 - 6x + 9 - 9) - 20 Step 5: Rewrite as: h(x) = -4[(x^2 - 6x + 9) - 9] - 20 Step 6: Factor the perfect square trinomial: h(x) = -4[(x - 3)^2 - 9] - 20 Step 7: Distribute the -4: h(x) = -4(x - 3)^2 + 36 - 20 Step 8: Simplify: h(x) = -4(x - 3)^2 + 16 Step 9: The vertex form is h(x) = -4(x - 3)^2 + 16, so the vertex is at (3, 16). Step 10: Since the coefficient of (x - 3)^2 is negative, the parabola opens downward, and the vertex represents the maximum point. Therefore, the maximum depth of the mirror is 16 cm, occurring at a horizontal distance of 3 cm from the left edge. The vertex is (3, 16).

5. Complete the square: x² + 9x + 15 = 0 → (x + h)² = k, find h and k? Answer: h = 9/2, k = 21/4 Solution: Start with x² + 9x + 15 = 0. Move the constant term to the right side: x² + 9x = -15. Take half of the coefficient of x (which is 9): 9/2.
   Full step-by-step solution

   Step 1: Start with x² + 9x + 15 = 0. Step 2: Move the constant term to the right side: x² + 9x = -15. Step 3: Take half of the coefficient of x (which is 9): 9/2. Step 4: Square it: (9/2)² = 81/4. Step 5: Add 81/4 to both sides: x² + 9x + 81/4 = -15 + 81/4. Step 6: The left side is a perfect square: (x + 9/2)² = -15 + 81/4. Step 7: Simplify the right side: -15 = -60/4, so -60/4 + 81/4 = 21/4. Step 8: Therefore, (x + 9/2)² = 21/4, so h = 9/2 and k = 21/4.

6. Mere is designing a parabolic reflecting pool for a new community garden. The depth of the pool (in centimeters) below the ground surface is modeled by the quadratic function d(x) = 4x^2 - 40x + 112, where x is the horizontal distance (in meters) from the left edge of the pool. Using the method of completing the square, rewrite the function in vertex form d(x) = a(x - p)^2 + q to determine the minimum depth of the pool (the deepest point) and the horizontal distance from the left edge where this minimum occurs. Then, state the coordinates of the vertex. Answer: Minimum depth of 12 centimeters at a horizontal distance of 5 meters; vertex at (5, 12) Solution: Start with the function: d(x) = 4x^2 - 40x + 112 Factor out 4 from the x^2 and x terms: d(x) = 4(x^2 - 10x) + 112 Complete the square inside the parentheses. Take half of -10, which is -5, and square it to get 25.
   Full step-by-step solution

   Step 1: Start with the function: d(x) = 4x^2 - 40x + 112 Step 2: Factor out 4 from the x^2 and x terms: d(x) = 4(x^2 - 10x) + 112 Step 3: Complete the square inside the parentheses. Take half of -10, which is -5, and square it to get 25. Step 4: Add and subtract 25 inside the parentheses: d(x) = 4(x^2 - 10x + 25 - 25) + 112 Step 5: Rewrite as: d(x) = 4[(x^2 - 10x + 25) - 25] + 112 Step 6: Factor the perfect square trinomial: d(x) = 4[(x - 5)^2 - 25] + 112 Step 7: Distribute the 4: d(x) = 4(x - 5)^2 - 100 + 112 Step 8: Simplify: d(x) = 4(x - 5)^2 + 12 Step 9: The vertex form is d(x) = 4(x - 5)^2 + 12, so the vertex is at (5, 12). Step 10: Since the coefficient of (x - 5)^2 is positive, the parabola opens upward, and the vertex represents the minimum point. Therefore, the minimum depth of the pool (deepest point) is 12 centimeters below the surface, occurring at a horizontal distance of 5 meters from the left edge. The vertex is (5, 12).

7. A rectangular courtyard is being designed in front of a community hall. The courtyard will have a decorative fountain in the center. The total area of the courtyard, in square meters, can be modeled by the quadratic function A(x) = 4x² - 24x + 52, where x is the distance in meters from the fountain to one edge of the courtyard. By completing the square, rewrite the area function in the vertex form A(x) = a(x - h)² + k, and then determine the minimum possible area of the courtyard. Answer: 16 Solution: Start with A(x) = 4x² - 24x + 52 Factor out the coefficient of x² from the x terms: A(x) = 4(x² - 6x) + 52 Complete the square inside the parentheses: take half of -6, which is -3, and square it to get 9 Add and subtract 9 inside the parentheses: A(x) = 4(x² - 6x + 9 - 9) + 52 Rewrite the…
   Full step-by-step solution

   Step 1: Start with A(x) = 4x² - 24x + 52 Step 2: Factor out the coefficient of x² from the x terms: A(x) = 4(x² - 6x) + 52 Step 3: Complete the square inside the parentheses: take half of -6, which is -3, and square it to get 9 Step 4: Add and subtract 9 inside the parentheses: A(x) = 4(x² - 6x + 9 - 9) + 52 Step 5: Rewrite the perfect square trinomial: A(x) = 4[(x - 3)² - 9] + 52 Step 6: Distribute the 4: A(x) = 4(x - 3)² - 36 + 52 Step 7: Combine constants: A(x) = 4(x - 3)² + 16 Step 8: The vertex form is A(x) = 4(x - 3)² + 16, so the vertex is at (3, 16). Since 4 > 0, the parabola opens upward, giving a minimum area. Step 9: The minimum possible area is the y-coordinate of the vertex: 16 square meters. The answer is 16.