Factor Quadratics

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Factor Quadratics · Grade 9 · Worksheet 3

1. Factor completely: 8x² + 34x + 21 = ? Answer: (4x + 3)(2x + 7) Solution: Multiply the leading coefficient (8) by the constant term (21): 8 × 21 = 168 Find two numbers that multiply to 168 and add to 34. The numbers are 6 and 28 because 6 × 28 = 168 and 6 + 28 = 34.
   Full step-by-step solution

   Step 1: Multiply the leading coefficient (8) by the constant term (21): 8 × 21 = 168 Step 2: Find two numbers that multiply to 168 and add to 34. The numbers are 6 and 28 because 6 × 28 = 168 and 6 + 28 = 34. Step 3: Rewrite the middle term using these numbers: 8x² + 6x + 28x + 21 Step 4: Factor by grouping: (8x² + 6x) + (28x + 21) Step 5: Factor out common factors from each group: 2x(4x + 3) + 7(4x + 3) Step 6: Factor out the common binomial (4x + 3): (4x + 3)(2x + 7) The completely factored form is (4x + 3)(2x + 7).

2. 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.

3. A physics class is designing a model rocket launch. The height of the rocket above ground is modeled by the function h(t) = -16t² + 96t + 112, where t is time in seconds. Factor this quadratic expression completely to determine at what time the rocket will hit the ground. Answer: 7 Solution: Set the height function equal to zero to find when the rocket hits the ground: -16t² + 96t + 112 = 0 Factor out the greatest common factor, which is -16: -16(t² - 6t - 7) = 0 Factor the quadratic expression inside the parentheses: t² - 6t - 7 = (t - 7)(t + 1) The complete factored form is: -16(t…
   Full step-by-step solution

   Step 1: Set the height function equal to zero to find when the rocket hits the ground: -16t² + 96t + 112 = 0 Step 2: Factor out the greatest common factor, which is -16: -16(t² - 6t - 7) = 0 Step 3: Factor the quadratic expression inside the parentheses: t² - 6t - 7 = (t - 7)(t + 1) Step 4: The complete factored form is: -16(t - 7)(t + 1) = 0 Step 5: Set each factor equal to zero: t - 7 = 0 or t + 1 = 0 Step 6: Solve for t: t = 7 or t = -1 Step 7: Since time cannot be negative, the rocket hits the ground at t = 7 seconds. The answer is 7.

4. A rectangular garden has an area represented by the expression 2x² + 7x + 6 square meters. If the length of the garden is (2x + 3) meters, what is the width of the garden in terms of x? Answer: (x + 2) Solution: Area = 2x² + 7x + 6 Length = (2x + 3) Width = ? Write the relationship between area, length, and width. Area = Length × Width So, Width = Area ÷ Length Substitute the expressions.
   Full step-by-step solution

   We are given: Area = 2x² + 7x + 6 Length = (2x + 3) Width = ? Step 1: Write the relationship between area, length, and width. Area = Length × Width So, Width = Area ÷ Length Step 2: Substitute the expressions. Width = (2x² + 7x + 6) ÷ (2x + 3) Step 3: Factor the quadratic expression 2x² + 7x + 6. We look for two numbers that multiply to (2 × 6) = 12 and add to 7. Those numbers are 3 and 4. Step 4: Rewrite the middle term using 3 and 4. 2x² + 7x + 6 = 2x² + 3x + 4x + 6 Step 5: Factor by grouping. Group the terms: (2x² + 3x) + (4x + 6) Factor each group: x(2x + 3) + 2(2x + 3) Step 6: Factor out the common factor (2x + 3). (2x + 3)(x + 2) So, 2x² + 7x + 6 = (2x + 3)(x + 2) Step 7: Now substitute back into the Width formula. Width = [(2x + 3)(x + 2)] ÷ (2x + 3) Step 8: Cancel the common factor (2x + 3) (assuming 2x + 3 ≠ 0). Width = x + 2 Final answer: The width is (x + 2) meters.

5. Factor completely: 6x² - 7x - 3 Answer: (3x + 1)(2x - 3) Solution: Identify coefficients. a = 6 b = -7 c = -3 Multiply a and c. a * c = 6 * (-3) = -18 Find two numbers that multiply to -18 and add to b (-7).
   Full step-by-step solution

   Let's factor 6x² - 7x - 3 completely. Step 1: Identify coefficients. The quadratic is in the form ax² + bx + c, where: a = 6 b = -7 c = -3 Step 2: Multiply a and c. a * c = 6 * (-3) = -18 Step 3: Find two numbers that multiply to -18 and add to b (-7). List factor pairs of -18: 1 and -18 → 1 + (-18) = -17 (no) -1 and 18 → -1 + 18 = 17 (no) 2 and -9 → 2 + (-9) = -7 (yes) -2 and 9 → -2 + 9 = 7 (no) 3 and -6 → 3 + (-6) = -3 (no) -3 and 6 → -3 + 6 = 3 (no) So the numbers are 2 and -9. Step 4: Rewrite the middle term (-7x) using these numbers. 6x² - 7x - 3 = 6x² + 2x - 9x - 3 Step 5: Factor by grouping. Group the first two terms and the last two terms: (6x² + 2x) + (-9x - 3) Factor out the greatest common factor from each group: From (6x² + 2x), factor out 2x: 2x(3x + 1) From (-9x - 3), factor out -3: -3(3x + 1) Now we have: 2x(3x + 1) - 3(3x + 1) Step 6: Factor out the common binomial factor (3x + 1). (3x + 1)(2x - 3) Step 7: Check by multiplying. (3x + 1)(2x - 3) = 3x * 2x + 3x * (-3) + 1 * 2x + 1 * (-3) = 6x² - 9x + 2x - 3 = 6x² - 7x - 3 ✓ Final answer: (3x + 1)(2x - 3)

6. A physics class is designing a projectile motion experiment where the height of a ball thrown upward is modeled by the function h(t) = -16t² + 64t + 80, where h is the height in feet and t is the time in seconds. The teacher asks students to factor this quadratic expression to determine at what time the ball will hit the ground. What is the factored form of the height function? Answer: -16(t - 5)(t + 1) Solution: Start with the quadratic function: h(t) = -16t² + 64t + 80 Factor out the greatest common factor, which is -16: h(t) = -16(t² - 4t - 5) Now factor the quadratic inside the parentheses: t² - 4t - 5 Find two numbers that multiply to -5 and add to -4: -5 and 1 Write the factored form: (t - 5)(t +…
   Full step-by-step solution

   Step 1: Start with the quadratic function: h(t) = -16t² + 64t + 80 Step 2: Factor out the greatest common factor, which is -16: h(t) = -16(t² - 4t - 5) Step 3: Now factor the quadratic inside the parentheses: t² - 4t - 5 Step 4: Find two numbers that multiply to -5 and add to -4: -5 and 1 Step 5: Write the factored form: (t - 5)(t + 1) Step 6: Include the -16 we factored out: h(t) = -16(t - 5)(t + 1) The factored form is -16(t - 5)(t + 1).

7. A right triangle is positioned on a coordinate plane with vertices at (0,0), (x,0), and (0,2x). The area of the triangle is 16 square units. Write the quadratic equation in standard form that represents the relationship between x and the given conditions. Answer: x^2 - 16 = 0 Solution: Identify the base and height of the triangle from the coordinates. The base is the distance from (0,0) to (x,0), which is x units. The height is the distance from (0,0) to (0,2x), which is 2x units.
   Full step-by-step solution

   Step 1: Identify the base and height of the triangle from the coordinates. The base is the distance from (0,0) to (x,0), which is x units. The height is the distance from (0,0) to (0,2x), which is 2x units. Step 2: Write the formula for the area of a right triangle: Area = (1/2) * base * height. Step 3: Substitute the given area and the expressions for base and height: 16 = (1/2) * x * (2x). Step 4: Simplify the equation: 16 = (1/2) * 2x^2 = x^2. Step 5: Rearrange into standard quadratic form: x^2 - 16 = 0. The quadratic equation in standard form is x^2 - 16 = 0.

8. 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.