Quadratic Formula

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Answer Key & Explanations

Quadratic Formula · Grade 9 · Worksheet 2

1. A rectangular garden has a length that is 5 meters more than its width. If the area of the garden is 84 square meters, what is the width of the garden in meters? Use the quadratic formula to solve. Answer: 7 Solution: Let the width = \( w \) meters. The length is 5 meters more than the width, so length = \( w + 5 \) meters.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width = \( w \) meters. The length is 5 meters more than the width, so length = \( w + 5 \) meters. --- **Step 2: Write the area equation** Area = length × width \( 84 = (w + 5) \times w \) --- **Step 3: Expand and rearrange into standard quadratic form** \( 84 = w^2 + 5w \) \( w^2 + 5w - 84 = 0 \) --- **Step 4: Identify coefficients for the quadratic formula** For \( a w^2 + b w + c = 0 \), \( a = 1 \), \( b = 5 \), \( c = -84 \). --- **Step 5: Write the quadratic formula** Quadratic formula: \( w = \frac{ -b \pm \sqrt{b^2 - 4ac} }{ 2a } \) --- **Step 6: Substitute values** \( w = \frac{ -5 \pm \sqrt{5^2 - 4(1)(-84)} }{ 2(1) } \) \( w = \frac{ -5 \pm \sqrt{25 + 336} }{ 2 } \) \( w = \frac{ -5 \pm \sqrt{361} }{ 2 } \) --- **Step 7: Simplify** \( \sqrt{361} = 19 \) So: \( w = \frac{ -5 + 19 }{ 2 } \) or \( w = \frac{ -5 - 19 }{ 2 } \) \( w = \frac{14}{2} \) or \( w = \frac{-24}{2} \) \( w = 7 \) or \( w = -12 \) --- **Step 8: Interpret the solutions** Width cannot be negative, so \( w = -12 \) is not valid. Thus, the width is \( 7 \) meters. --- **Final answer:** 7

2. A company's profit P (in thousands of dollars) from selling x units of a product is modeled by the quadratic equation P = -2x² + 120x - 1000. How many units must the company sell to break even (make zero profit)? Answer: 10 Solution: Set the profit equation equal to zero to find break-even point: -2x² + 120x - 1000 = 0 Divide the entire equation by -2 to simplify: x² - 60x + 500 = 0 Use the quadratic formula x = [-b ± sqrt(b² - 4ac)] / (2a) where a = 1, b = -60, c = 500 Calculate discriminant: (-60)² - 4(1)(500) = 3600 -…
   Full step-by-step solution

   Step 1: Set the profit equation equal to zero to find break-even point: -2x² + 120x - 1000 = 0 Step 2: Divide the entire equation by -2 to simplify: x² - 60x + 500 = 0 Step 3: Use the quadratic formula x = [-b ± sqrt(b² - 4ac)] / (2a) where a = 1, b = -60, c = 500 Step 4: Calculate discriminant: (-60)² - 4(1)(500) = 3600 - 2000 = 1600 Step 5: Calculate square root: sqrt(1600) = 40 Step 6: Apply quadratic formula: x = [60 ± 40] / 2 Step 7: Calculate both solutions: x = (60 + 40)/2 = 100/2 = 50, and x = (60 - 40)/2 = 20/2 = 10 Step 8: The company breaks even at 10 units (the lower quantity where profit first reaches zero) and again at 50 units (where profit returns to zero after peaking). The answer is 10 units.

3. Mason is constructing a parabolic water fountain for a school project. The height of the water stream, in meters, is given by the equation h(d) = -3d² + 12d + 7, where d is the horizontal distance from the nozzle in meters. At what horizontal distance(s) from the nozzle will the water stream be exactly 7 meters high? Answer: 0 meters and 4 meters Solution: Set the height equation equal to 7: -3d² + 12d + 7 = 7. Subtract 7 from both sides: -3d² + 12d + 7 - 7 = 0 → -3d² + 12d = 0.
   Full step-by-step solution

   Step 1: Set the height equation equal to 7: -3d² + 12d + 7 = 7. Step 2: Subtract 7 from both sides: -3d² + 12d + 7 - 7 = 0 → -3d² + 12d = 0. Step 3: Factor out the greatest common factor: 3d(-d + 4) = 0 (or multiply both sides by -1 to get 3d² - 12d = 0, then factor: 3d(d - 4) = 0). Step 4: Set each factor equal to zero: 3d = 0 or -d + 4 = 0 → d = 0 or d = 4. Step 5: The water stream is exactly 7 meters high at d = 0 meters (at the nozzle) and d = 4 meters from the nozzle. The answer is 0 meters and 4 meters.

4. A parabolic bridge is drawn on a coordinate plane with its vertex at the point (0, 7). The bridge intersects the x-axis at points (-7, 0) and (7, 0). Using the quadratic formula, determine the x-coordinates of the points where the bridge's arch is at a height of 3 units above the x-axis. Write the quadratic equation in standard form that models the arch, then solve for the x-coordinates. Answer: x = ±(2√7)/3 Solution: The vertex is at (0, 7), so the equation in vertex form is y = a(x - 0)^2 + 7, or y = ax^2 + 7. The arch passes through (7, 0), so substitute x = 7 and y = 0: 0 = a(7)^2 + 7 → 0 = 49a + 7 → 49a = -7 → a = -7/49 = -1/7.
   Full step-by-step solution

   Step 1: The vertex is at (0, 7), so the equation in vertex form is y = a(x - 0)^2 + 7, or y = ax^2 + 7. Step 2: The arch passes through (7, 0), so substitute x = 7 and y = 0: 0 = a(7)^2 + 7 → 0 = 49a + 7 → 49a = -7 → a = -7/49 = -1/7. Step 3: The equation of the arch is y = (-1/7)x^2 + 7. Step 4: To find where the height is 3 units, set y = 3: 3 = (-1/7)x^2 + 7. Step 5: Subtract 7 from both sides: -4 = (-1/7)x^2. Step 6: Multiply both sides by -7: 28 = x^2. Step 7: Rewrite as a quadratic equation in standard form: x^2 - 28 = 0, so a = 1, b = 0, c = -28. Step 8: Apply the quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / (2a) → x = [0 ± sqrt(0^2 - 4(1)(-28))] / (2(1)). Step 9: Simplify inside the square root: x = [± sqrt(112)] / 2. Step 10: Simplify sqrt(112) = sqrt(16 * 7) = 4*sqrt(7). Step 11: So x = [± 4*sqrt(7)] / 2 = ± 2*sqrt(7). The x-coordinates where the arch is at a height of 3 units are x = 2√7 and x = -2√7.

5. 3x² - 11x + 8 = 0 Answer: x = 2.667, x = 1 Solution: Identify coefficients: a = 3, b = -11, c = 8 Write the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a) Substitute the values: x = [11 ± √((-11)² - 4*3*8)] / (2*3) Calculate the discriminant: (-11)² - 4*3*8 = 121 - 96 = 25 Take the square root: √25 = 5 Apply the formula: x = [11 ± 5] / 6 x = (11…
   Full step-by-step solution

   Step 1: Identify coefficients: a = 3, b = -11, c = 8 Step 2: Write the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a) Step 3: Substitute the values: x = [11 ± √((-11)² - 4*3*8)] / (2*3) Step 4: Calculate the discriminant: (-11)² - 4*3*8 = 121 - 96 = 25 Step 5: Take the square root: √25 = 5 Step 6: Apply the formula: x = [11 ± 5] / 6 Step 7: Calculate the two solutions: x = (11 + 5) / 6 = 16 / 6 = 8/3 = 2.667 x = (11 - 5) / 6 = 6 / 6 = 1 Step 8: The solutions are x = 8/3 (or 2.667) and x = 1

6. Noah is designing a parabolic water fountain that will be modeled on a coordinate grid. The vertex of the fountain's arc is at (0, 6), and the water stream hits the ground at points (−1, 0) and (x, 0). Use the quadratic formula to find the quadratic equation in standard form that models the fountain's arc, then determine the positive x-coordinate where the stream lands. Answer: 1 Solution: The vertex is at (0, 6), so vertex form is y = a(x - 0)^2 + 6 = ax^2 + 6. The stream passes through (-1, 0). Substitute: 0 = a(-1)^2 + 6 → 0 = a + 6 → a = -6.
   Full step-by-step solution

   Step 1: The vertex is at (0, 6), so vertex form is y = a(x - 0)^2 + 6 = ax^2 + 6. Step 2: The stream passes through (-1, 0). Substitute: 0 = a(-1)^2 + 6 → 0 = a + 6 → a = -6. Step 3: The equation is y = -6x^2 + 6. In standard form: -6x^2 + 0x + 6 = 0. Step 4: Identify a = -6, b = 0, c = 6. Quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / (2a). Step 5: Substitute: x = [0 ± sqrt(0^2 - 4(-6)(6))] / (2(-6)) = [± sqrt(144)] / (-12) = ±12 / (-12). Step 6: x = 12/(-12) = -1, or x = -12/(-12) = 1. Step 7: The positive x-coordinate is 1. The answer is 1.

7. A parabolic arch is constructed with its vertex at the origin (0,0) of a coordinate plane. The arch spans 24 meters wide at its base, with the base endpoints at (-12,0) and (12,0). If the arch reaches a maximum height of 9 meters, write the quadratic equation in standard form that models this arch, then determine the height of the arch at a point 6 meters from the center. Answer: 6.75 Solution: Since the vertex is at (0,0) and the parabola opens downward, the equation is y = ax^2. Use the point (12,-9) because the arch spans from (-12,0) to (12,0) with maximum height 9, meaning at x=12, y=-9.
   Full step-by-step solution

   Step 1: Since the vertex is at (0,0) and the parabola opens downward, the equation is y = ax^2. Step 2: Use the point (12,-9) because the arch spans from (-12,0) to (12,0) with maximum height 9, meaning at x=12, y=-9. Step 3: Substitute into the equation: -9 = a(12)^2 Step 4: Calculate: -9 = 144a Step 5: Solve for a: a = -9/144 = -1/16 Step 6: The equation is y = (-1/16)x^2 Step 7: To find height at 6 meters from center, substitute x=6: y = (-1/16)(6)^2 Step 8: Calculate: y = (-1/16)(36) = -36/16 = -9/4 = -2.25 Step 9: Since the question asks for height, and we measured from the top, the height is 9 - 2.25 = 6.75 meters Step 10: The height of the arch 6 meters from the center is 6.75 meters.