Quadratic Formula

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Answer Key & Explanations

Quadratic Formula · Grade 9 · Worksheet 3

1. Liam is deriving the quadratic formula by completing the square on the general quadratic equation ax² + bx + c = 0, where a ≠ 0. He visualizes the process as a geometric area model: he starts with a large square of side length x representing the x² term, then adds a rectangle of area bx (where b = 10) along one side, and finally includes a constant area c = 25. He wants to complete the square by adding a small square to form a perfect square trinomial. Show the step-by-step derivation of the quadratic formula x = [-b ± √(b² - 4ac)] / (2a) from ax² + bx + c = 0 using the method of completing the square, using the specific numbers a = 5, b = 10, c = 25 as a concrete example to illustrate each step, then generalize. Answer: x = [-10 ± √(100 - 500)] / 10 Solution: Start with the general quadratic equation: ax² + bx + c = 0. Subtract c from both sides: ax² + bx = -c. Divide both sides by a (a ≠ 0): x² + (b/a)x = -c/a.
   Full step-by-step solution

   Step 1: Start with the general quadratic equation: ax² + bx + c = 0. Step 2: Subtract c from both sides: ax² + bx = -c. Step 3: Divide both sides by a (a ≠ 0): x² + (b/a)x = -c/a. Step 4: Complete the square. Take half of the coefficient of x: (b/(2a)). Square it: (b/(2a))² = b²/(4a²). Step 5: Add this term to both sides: x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²). Step 6: The left side is a perfect square: (x + b/(2a))² = -c/a + b²/(4a²). Step 7: Combine the right side over a common denominator: (x + b/(2a))² = (b² - 4ac) / (4a²). Step 8: Take the square root of both sides: x + b/(2a) = ± √(b² - 4ac) / (2a). Step 9: Isolate x: x = -b/(2a) ± √(b² - 4ac) / (2a). Step 10: Combine the fractions: x = [-b ± √(b² - 4ac)] / (2a). Now substitute a = 5, b = 10, c = 25: Step 11: x = [-10 ± √(10² - 4·5·25)] / (2·5). Step 12: x = [-10 ± √(100 - 500)] / 10. Step 13: x = [-10 ± √(-400)] / 10, which gives complex roots. Thus the derivation yields the quadratic formula.

2. x² + 5x + 6 = 0 Answer: x = -2, -3 Solution: x² + 5x + 6 = 0 Identify the coefficients. The equation is in the form ax² + bx + c = 0. Here, a = 1, b = 5, c = 6.
   Full step-by-step solution

   We are solving the quadratic equation: x² + 5x + 6 = 0 Step 1: Identify the coefficients. The equation is in the form ax² + bx + c = 0. Here, a = 1, b = 5, c = 6. Step 2: Factor the quadratic expression. We look for two numbers that multiply to a * c = 1 * 6 = 6, and add to b = 5. The numbers 2 and 3 multiply to 6 and add to 5. Step 3: Rewrite the middle term using these numbers. x² + 5x + 6 = x² + 2x + 3x + 6. Step 4: Factor by grouping. Group the terms: (x² + 2x) + (3x + 6). Factor each group: x(x + 2) + 3(x + 2). Step 5: Factor out the common binomial (x + 2). (x + 2)(x + 3) = 0. Step 6: Apply the zero-product property. If (x + 2)(x + 3) = 0, then: x + 2 = 0 or x + 3 = 0. Step 7: Solve each equation. x + 2 = 0 → x = -2 x + 3 = 0 → x = -3 Final answer: x = -2, -3

3. Tane is a mathematician working on a new model for projectile motion. He starts with the general quadratic equation in standard form: ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. Tane wants to derive the quadratic formula, x = [-b ± sqrt(b² - 4ac)] / (2a), by completing the square on this general equation. Show all the steps Tane must follow to complete this derivation. Answer: x = [-b ± sqrt(b² - 4ac)] / (2a) Solution: Start with the general quadratic equation: ax² + bx + c = 0, where a ≠ 0. Subtract c from both sides to isolate the x-terms: ax² + bx = -c.
   Full step-by-step solution

   Step 1: Start with the general quadratic equation: ax² + bx + c = 0, where a ≠ 0. Step 2: Subtract c from both sides to isolate the x-terms: ax² + bx = -c. Step 3: Divide every term by a to make the coefficient of x² equal to 1: x² + (b/a)x = -c/a. Step 4: Complete the square on the left side. Take half of the coefficient of x, which is (b/a) / 2 = b/(2a), then square it: [b/(2a)]² = b²/(4a²). Step 5: Add this term to both sides of the equation: x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²). Step 6: The left side is now a perfect square trinomial. Factor it as (x + b/(2a))²: (x + b/(2a))² = -c/a + b²/(4a²). Step 7: Combine the terms on the right side by finding a common denominator of 4a²: -c/a = -4ac/(4a²). So, (x + b/(2a))² = (-4ac + b²) / (4a²) = (b² - 4ac) / (4a²). Step 8: Take the square root of both sides. Remember to include both positive and negative square roots: x + b/(2a) = ± sqrt[(b² - 4ac) / (4a²)]. Step 9: Simplify the square root on the right side: sqrt[(b² - 4ac) / (4a²)] = sqrt(b² - 4ac) / sqrt(4a²) = sqrt(b² - 4ac) / (2a). Note that sqrt(4a²) = 2|a|, but since we have ±, we can write ± sqrt(b² - 4ac) / (2a). Step 10: Solve for x by subtracting b/(2a) from both sides: x = -b/(2a) ± sqrt(b² - 4ac) / (2a). Step 11: Combine the terms over the common denominator 2a: x = [-b ± sqrt(b² - 4ac)] / (2a). This is the quadratic formula. Tane can now use this formula to find the x-intercepts of any quadratic equation.

4. Mere is a structural engineer designing a rectangular solar panel array. The array's length is 6 meters more than twice its width. The area of the array is 216 square meters. Set up a quadratic equation in standard form (ax² + bx + c = 0) to model this situation. Then, derive the quadratic formula from the general form ax² + bx + c = 0 by completing the square, and use it to find the width of the solar panel array. Answer: x = 9 Solution: Let the width be x meters. The length is 2x + 6 meters. Area = length × width = (2x + 6) * x = 2x² + 6x = 216.
   Full step-by-step solution

   Step 1: Let the width be x meters. Step 2: The length is 2x + 6 meters. Step 3: Area = length × width = (2x + 6) * x = 2x² + 6x = 216. Step 4: Rearrange to standard form: 2x² + 6x - 216 = 0. Step 5: Divide by 2: x² + 3x - 108 = 0. Here a = 1, b = 3, c = -108. Step 6: Derive the quadratic formula from ax² + bx + c = 0: a) Divide by a: x² + (b/a)x + c/a = 0 b) Move constant: x² + (b/a)x = -c/a c) Complete the square: x² + (b/a)x + (b/(2a))² = -c/a + (b/(2a))² d) Factor left: (x + b/(2a))² = (b² - 4ac)/(4a²) e) Take square root: x + b/(2a) = ± sqrt(b² - 4ac)/(2a) f) Solve for x: x = [-b ± sqrt(b² - 4ac)] / (2a) Step 7: Substitute a = 1, b = 3, c = -108: x = [-3 ± sqrt(3² - 4(1)(-108))] / (2(1)) x = [-3 ± sqrt(9 + 432)] / 2 x = [-3 ± sqrt(441)] / 2 x = [-3 ± 21] / 2 Step 8: Two solutions: x = (-3 + 21)/2 = 18/2 = 9 x = (-3 - 21)/2 = -24/2 = -12 (reject, width cannot be negative) Step 9: The width is 9 meters.

5. Sophia is designing a rectangular garden for a community project. The length of the garden is 11 meters more than its width, and the area of the garden is 126 square meters. To find the width of the garden, she needs to derive the quadratic formula from the general quadratic equation ax² + bx + c = 0 by completing the square. Starting with the general form, show the complete derivation to obtain the quadratic formula x = [-b ± √(b² - 4ac)] / (2a). Answer: x = [-b ± √(b² - 4ac)] / (2a) Solution: Start with the general quadratic equation in standard form: ax² + bx + c = 0, where a ≠ 0. Subtract c from both sides to isolate the x terms: ax² + bx = -c.
   Full step-by-step solution

   Step 1: Start with the general quadratic equation in standard form: ax² + bx + c = 0, where a ≠ 0. Step 2: Subtract c from both sides to isolate the x terms: ax² + bx = -c. Step 3: Divide both sides by a (since a ≠ 0) to make the coefficient of x² equal to 1: x² + (b/a)x = -c/a. Step 4: Complete the square. Take half of the coefficient of x, which is (b/a)/2 = b/(2a). Square it: (b/(2a))² = b²/(4a²). Add this to both sides of the equation: x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²). Step 5: The left side is now a perfect square trinomial. Factor it as (x + b/(2a))². The right side becomes: b²/(4a²) - c/a. To combine the terms on the right, write c/a with a denominator of 4a²: c/a = (4ac)/(4a²). So the right side is: (b² - 4ac)/(4a²). Step 6: Now the equation is: (x + b/(2a))² = (b² - 4ac)/(4a²). Step 7: Take the square root of both sides. Remember to include both the positive and negative square roots: x + b/(2a) = ± √(b² - 4ac) / (2a). Step 8: Subtract b/(2a) from both sides to solve for x: x = -b/(2a) ± √(b² - 4ac) / (2a). Step 9: Combine the terms over the common denominator 2a: x = [-b ± √(b² - 4ac)] / (2a). The quadratic formula is derived: x = [-b ± √(b² - 4ac)] / (2a).

6. Derive the quadratic formula x = [-b ± √(b²-4ac)] / (2a) by completing the square on ax² + bx + c = 0. Answer: x = [-b ± √(b²-4ac)] / (2a) Solution: Start with the general quadratic equation: ax² + bx + c = 0 Subtract c from both sides: ax² + bx = -c Divide both sides by a (a ≠ 0): x² + (b/a)x = -c/a Complete the square.
   Full step-by-step solution

   Step 1: Start with the general quadratic equation: ax² + bx + c = 0 Step 2: Subtract c from both sides: ax² + bx = -c Step 3: Divide both sides by a (a ≠ 0): x² + (b/a)x = -c/a Step 4: Complete the square. Take half of the coefficient of x: (b/a) ÷ 2 = b/(2a). Square it: (b/(2a))² = b²/(4a²). Add this to both sides: x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²) Step 5: Write the left side as a perfect square: (x + b/(2a))² = -c/a + b²/(4a²) Step 6: Combine the right side over a common denominator 4a²: (x + b/(2a))² = (-4ac + b²)/(4a²) = (b² - 4ac)/(4a²) Step 7: Take the square root of both sides: x + b/(2a) = ±√(b² - 4ac)/(2a) Step 8: Isolate x by subtracting b/(2a) from both sides: x = -b/(2a) ± √(b² - 4ac)/(2a) Step 9: Combine the terms over the common denominator 2a: x = [-b ± √(b² - 4ac)]/(2a) The quadratic formula is derived as x = [-b ± √(b²-4ac)]/(2a).