Factor Quadratics

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Factor Quadratics · Grade 9 · Worksheet 3

1. A physics student is analyzing the trajectory of a rocket launched from ground level. The height of the rocket above the ground is modeled by the quadratic function h(t) = -5t² + 40t, where h is the height in meters and t is the time in seconds after launch. Factor this quadratic expression to determine at what time the rocket will return to the ground. Answer: 8 Solution: Set the height function equal to zero since the rocket returns to the ground when height is 0: -5t² + 40t = 0 Factor out the greatest common factor, which is 5t: 5t(-t + 8) = 0 Alternatively, we can write this as: -5t(t - 8) = 0 Set each factor equal to zero: -5t = 0 or (t - 8) = 0 Solve each…
   Full step-by-step solution

   Step 1: Set the height function equal to zero since the rocket returns to the ground when height is 0: -5t² + 40t = 0 Step 2: Factor out the greatest common factor, which is 5t: 5t(-t + 8) = 0 Step 3: Alternatively, we can write this as: -5t(t - 8) = 0 Step 4: Set each factor equal to zero: -5t = 0 or (t - 8) = 0 Step 5: Solve each equation: t = 0 or t = 8 Step 6: t = 0 represents the launch time, so t = 8 represents when the rocket returns to the ground The answer is 8 seconds.

2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,x+3). The area of the triangle is 20 square units. Write the quadratic equation in standard form that represents the relationship between x and the given conditions. Answer: x² + 3x - 40 = 0 Solution: - Base: distance between (0,0) and (x,0) = x units - Height: distance between (0,0) and (0,x+3) = x+3 units Area = (1/2) × base × height 20 = (1/2) × x × (x+3) Multiply both sides by 2 to eliminate the fraction 40 = x × (x+3) 40 = x² + 3x x² + 3x - 40 = 0 The quadratic equation in standard form…
   Full step-by-step solution

   Step 1: Identify the base and height from the coordinates - Base: distance between (0,0) and (x,0) = x units - Height: distance between (0,0) and (0,x+3) = x+3 units Step 2: Apply the area formula for triangles Area = (1/2) × base × height 20 = (1/2) × x × (x+3) Step 3: Multiply both sides by 2 to eliminate the fraction 40 = x × (x+3) Step 4: Expand the right side 40 = x² + 3x Step 5: Write in standard quadratic form x² + 3x - 40 = 0 The quadratic equation in standard form is x² + 3x - 40 = 0.

3. Factor completely: 8x² - 22x + 12 = ? Answer: 2(4x - 3)(x - 2) Solution: Factor out the greatest common factor (GCF) of 2: 8x² - 22x + 12 = 2(4x² - 11x + 6). Factor the quadratic 4x² - 11x + 6. Multiply the leading coefficient (4) by the constant term (6): 4 × 6 = 24.
   Full step-by-step solution

   Step 1: Factor out the greatest common factor (GCF) of 2: 8x² - 22x + 12 = 2(4x² - 11x + 6). Step 2: Factor the quadratic 4x² - 11x + 6. Multiply the leading coefficient (4) by the constant term (6): 4 × 6 = 24. Step 3: Find two numbers that multiply to 24 and add to -11: -3 and -8. Step 4: Rewrite the middle term using these numbers: 4x² - 3x - 8x + 6. Step 5: Factor by grouping: (4x² - 3x) + (-8x + 6) = x(4x - 3) - 2(4x - 3). Step 6: Factor out the common binomial (4x - 3): (4x - 3)(x - 2). Step 7: Include the GCF: 2(4x - 3)(x - 2). The completely factored form is 2(4x - 3)(x - 2).

4. Aroha is designing a rectangular mural for her school's art exhibition. The area of the mural is given by the quadratic expression 9x² + 33x + 30 square meters, where x represents the number of additional meters beyond the minimum width. The length of the mural is (3x + 5) meters. What expression in factored form represents the width of the mural? Answer: (3x + 6) Solution: Area of a rectangle = length * width, so width = area / length. Width = (9x² + 33x + 30) / (3x + 5). Factor the numerator 9x² + 33x + 30.
   Full step-by-step solution

   Step 1: Area of a rectangle = length * width, so width = area / length. Step 2: Width = (9x² + 33x + 30) / (3x + 5). Step 3: Factor the numerator 9x² + 33x + 30. Step 4: Multiply the coefficient of x² (9) by the constant term (30): 9 * 30 = 270. Step 5: Find two numbers that multiply to 270 and add to 33. The numbers are 18 and 15 (since 18 * 15 = 270 and 18 + 15 = 33). Step 6: Rewrite the middle term using these numbers: 9x² + 18x + 15x + 30. Step 7: Group the terms: (9x² + 18x) + (15x + 30). Step 8: Factor each group: 9x(x + 2) + 15(x + 2). Step 9: Factor out the common binomial (x + 2): (x + 2)(9x + 15). Step 10: Further factor the second binomial: 9x + 15 = 3(3x + 5). So (x + 2)(3)(3x + 5) = 3(x + 2)(3x + 5). Step 11: Width = 3(x + 2)(3x + 5) / (3x + 5) = 3(x + 2) = 3x + 6. The width of the mural is (3x + 6) meters.

5. Factor completely: 10x² + 29x - 21 = ? Answer: (5x - 3)(2x + 7) Solution: Multiply the leading coefficient (10) by the constant term (-21): 10 × (-21) = -210. Find two numbers that multiply to -210 and add to 29. The numbers are 35 and -6 because 35 × (-6) = -210 and 35 + (-6) = 29.
   Full step-by-step solution

   Step 1: Multiply the leading coefficient (10) by the constant term (-21): 10 × (-21) = -210. Step 2: Find two numbers that multiply to -210 and add to 29. The numbers are 35 and -6 because 35 × (-6) = -210 and 35 + (-6) = 29. Step 3: Rewrite the middle term 29x as 35x - 6x: 10x² + 35x - 6x - 21. Step 4: Factor by grouping: (10x² + 35x) + (-6x - 21). Step 5: Factor out the greatest common factor from each group: 5x(2x + 7) - 3(2x + 7). Step 6: Factor out the common binomial (2x + 7): (5x - 3)(2x + 7). The completely factored form is (5x - 3)(2x + 7).

6. 2x² + 7x + 6 = ? Answer: (2x + 3)(x + 2) Solution: When factoring quadratic expressions of the form ax² + bx + c, we use the AC method where we find two numbers that multiply to a×c and add to b. Then we rewrite the middle term using these numbers and factor by grouping.
7. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,6). The area of the triangle is 12 square units. What is the value of x? Answer: 4 Solution: The vertices are at (0,0), (x,0), and (0,6). This means the triangle has legs along the x-axis and y-axis, so it is a right triangle with the right angle at (0,0). Identify the base and height.
   Full step-by-step solution

   Step 1: Understand the triangle's vertices. The vertices are at (0,0), (x,0), and (0,6). This means the triangle has legs along the x-axis and y-axis, so it is a right triangle with the right angle at (0,0). Step 2: Identify the base and height. The base is along the x-axis from (0,0) to (x,0), so the base length is |x|. The height is along the y-axis from (0,0) to (0,6), so the height is 6. Step 3: Write the formula for the area of a triangle. Area = (1/2) * base * height. Step 4: Substitute known values into the formula. Area = 12, base = x, height = 6. So: 12 = (1/2) * x * 6. Step 5: Simplify the equation. First, (1/2) * 6 = 3. So: 12 = 3 * x. Step 6: Solve for x. Divide both sides by 3: x = 12 / 3. x = 4. Step 7: Conclusion. The value of x is 4.

8. Sophia is designing a rectangular solar panel for a school project. The area of the solar panel is given by the quadratic expression 6x² + 19x + 10 square centimeters. The length of the panel is (3x + 2) centimeters. What expression in factored form represents the width of the solar panel? Answer: (2x + 5) Solution: The area of a rectangle is length times width, so width = area / length. Width = (6x² + 19x + 10) / (3x + 2). Factor the numerator 6x² + 19x + 10.
   Full step-by-step solution

   Step 1: The area of a rectangle is length times width, so width = area / length. Step 2: Width = (6x² + 19x + 10) / (3x + 2). Step 3: Factor the numerator 6x² + 19x + 10. Step 4: Find two numbers that multiply to 6 * 10 = 60 and add to 19. The numbers are 15 and 4. Step 5: Rewrite the middle term: 6x² + 15x + 4x + 10. Step 6: Group terms: (6x² + 15x) + (4x + 10). Step 7: Factor each group: 3x(2x + 5) + 2(2x + 5). Step 8: Factor out the common binomial (2x + 5): (2x + 5)(3x + 2). Step 9: Width = (2x + 5)(3x + 2) / (3x + 2) = 2x + 5. The width is (2x + 5) centimeters.