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Mastering Quadratic Equations by Factoring: Examples and Solutions

Explore how to solve quadratic equations by factoring with detailed examples and step-by-step solutions. Verify answers using the Zero Product Property. Practice identifying solutions and checking them for accuracy.

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Mastering Quadratic Equations by Factoring: Examples and Solutions

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  1. Warm Up • Find each product. • 1. (x + 2)(x + 7) 2. (x – 11)(x + 5) • 3. (x – 10)2 • Factor each polynomial. • 4. x2 + 12x + 35 5. x2 + 2x – 63 • 6. x2 – 10x + 16 7. 2x2 – 16x + 32 x2 + 9x + 14 x2 – 6x – 55 x2 – 20x + 100 (x + 5)(x + 7) (x– 7)(x + 9) (x – 2)(x – 8) 2(x – 4)2

  2. 9-6 Solving Quadratic Equations by Factoring Holt Algebra 1

  3. Example 1A: Use the Zero Product Property Use the Zero Product Property to solve the equation. Check your answer. (x – 7)(x + 2) = 0 Use the Zero Product Property. x – 7 = 0 or x + 2 = 0 Solve each equation. x = 7 or x = –2 The solutions are 7 and –2.

  4. Check(x – 7)(x + 2) = 0 (–2 – 7)(–2+ 2) 0 (–9)(0) 0  0 0 Check(x – 7)(x + 2) = 0 (7 – 7)(7+ 2) 0 (0)(9) 0  0 0 Example 1A Continued Use the Zero Product Property to solve the equation. Check your answer. Substitute each solution for x into the original equation.

  5. Check (x – 2)(x) = 0 (x – 2)(x) = 0 (0 – 2)(0) 0 (2 – 2)(2) 0 (0)(2) 0 (–2)(0) 0   0 0 0 0 Example 1B: Use the Zero Product Property Use the Zero Product Property to solve each equation. Check your answer. (x – 2)(x) = 0 (x)(x – 2) = 0 Use the Zero Product Property. x= 0 or x – 2 = 0 Solve the second equation. x = 2 The solutions are 0 and 2. Substitute each solution for x into the original equation.

  6. x = 4 or x = 2 The solutions are 4 and 2. Check Check x2 – 6x + 8 = 0 x2 – 6x + 8 = 0 (4)2 – 6(4) + 8 0 (2)2 – 6(2) + 8 0 16 – 24 + 8 0 4 – 12 + 8 0   0 0 0 0 Example 2A: Solving Quadratic Equations by Factoring Solve the quadratic equation by factoring. Check your answer. x2 – 6x + 8 = 0 (x – 4)(x – 2) = 0 Factor the trinomial. x – 4 = 0 or x – 2 = 0 Use the Zero Product Property. Solve each equation.

  7. x2 + 4x = 21 –21 –21 x2 + 4x – 21 = 0 x = –7 or x =3 The solutions are –7 and 3. Example 2B: Solving Quadratic Equations by Factoring Solve the quadratic equation by factoring. Check your answer. x2 + 4x = 21 The equation must be written in standard form. So subtract 21 from both sides. (x + 7)(x –3) = 0 Factor the trinomial. x + 7 = 0 or x – 3 = 0 Use the Zero Product Property. Solve each equation.

  8. –2x2 = 20x + 50 +2x2 +2x2 0 = 2x2 + 20x + 50 Example 2D: Solving Quadratic Equations by Factoring Solve the quadratic equation by factoring. Check your answer. –2x2 = 20x + 50 The equation must be written in standard form. So add 2x2 to both sides. 2x2 + 20x + 50 = 0 Factor out the GCF 2. 2(x2 + 10x + 25) = 0 Factor the trinomial. 2(x + 5)(x + 5) = 0 2 ≠ 0 or x + 5 = 0 Use the Zero Product Property. x = –5 Solve the equation.

  9. Check It Out! Example 2c Solve the quadratic equation by factoring. Check your answer. 30x = –9x2 – 25 Write the equation in standard form. –9x2 – 30x – 25 = 0 –1(9x2 + 30x + 25) = 0 Factor out the GCF, –1. –1(3x + 5)(3x + 5) = 0 Factor the trinomial. –1 ≠ 0 or 3x + 5 = 0 Use the Zero Product Property. – 1 cannot equal 0. Solve the remaining equation.

  10. Check It Out! Example 3 What if…? The equation for the height above the water for another diver can be modeled by h = –16t2 + 8t + 24. Find the time it takes this diver to reach the water. h = –16t2 + 8t + 24 The diver reaches the water when h = 0. 0 = –16t2 + 8t + 24 0 = –8(2t2 – t – 3) Factor out the GFC, –8. 0 = –8(2t – 3)(t + 1) Factor the trinomial.

  11. 0 –16(1.5)2 + 8(1.5) + 24 0 –36 + 12 + 24 0 0 Check It Out! Example 3 Continued Use the Zero Product Property. –8 ≠0, 2t – 3 = 0 or t + 1= 0  2t = 3 or t = –1 Solve each equation. Since time cannot be negative, –1 does not make sense in this situation. t = 1.5 It takes the diver 1.5 seconds to reach the water. Check 0 = –16t2 + 8t + 24 Substitute 1 into the original equation. 

  12. Lesson Quiz: Part I • Use the Zero Product Property to solve each equation. Check your answers. • 1. (x – 10)(x + 5) = 0 • 2. (x + 5)(x) = 0 • Solve each quadratic equation by factoring. Check your answer. • 3. x2 + 16x + 48 = 0 • 4. x2 – 11x = –24 10, –5 –5, 0 –4, –12 3, 8

  13. Lesson Quiz: Part II 1, –7 5. 2x2 + 12x – 14 = 0 6. x2 + 18x + 81 = 0 –9 –2 7. –4x2 = 16x + 16 8. The height of a rocket launched upward from a 160 foot cliff is modeled by the function h(t) = –16t2 + 48t + 160, where h is height in feet and t is time in seconds. Find the time it takes the rocket to reach the ground at the bottom of the cliff. 5 s

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