Zero product property notes
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Using the zero product property, this document provides examples of solving equations by factorizing expressions with two factors and setting each factor equal to zero. It demonstrates solving equations of the form (factor 1)(factor 2)=0 for various variable expressions including linear, quadratic, and cubic equations. Step-by-step solutions are shown along with the process of factorizing, setting factors equal to zero, solving the resulting simple equations, and stating the solution sets.
Zero product property notes
- 1. Using the zero product property to solve equations once you have factored.
- 2. Zero Product Property If a • b = 0 then a=0, b=0, or both a and b equal 0.
- 3. 1. Solve (x + 3)(x - 5) = 0 Using the Zero Product Property, you know that either x + 3 = 0 or x - 5 =0 Solve each equation. x = -3 or x = 5 {-3, 5}
- 4. 2. Solve (2a + 4)(a + 7) = 0 2a + 4 = 0 or a + 7 = 0 2a = -4 or a = -7 a = -2 or a = -7 {-2, -7}
- 5. 3. Solve (3t + 5)(t - 3) = 0 3t + 5 = 0 or t - 3 = 0 3t = -5 or t = 3 t = -5/3 or t = 3 {-5/3, 3}
- 6. Solve (y – 3)(2y + 6) = 0 1. {-3, 3} 2. {-3, 6} 3. {3, 6} 4. {3, -6}
- 7. 4 steps for solving a quadratic equation 1. Set the equation equal to 0. Set = 0 2. Factor the equation. Factor Split/Solve 3. Set each part equal to 0 and Check solve. 4. Check your answer on the calculator.
- 8. 4. Solve x2 - 11x = 0 GCF = x Set = 0 x(x - 11) = 0 Factor Split/Solve x = 0 or x - 11 = 0 Check x = 0 or x = 11 {0, 11}
- 9. 5. Solve. -24a +144 = -a2 Put it in descending order. Set = 0 a2 - 24a + 144 = 0 Factor Split/Solve (a - 12)2 = 0 Check a - 12 = 0 a = 12 {12}
- 10. 6. Solve 4m2 + 25 = 20m 4m2 - 20m + 25 = 0 Set = 0 (2m - 5)2 = 0 Factor Split/Solve 2m - 5 = 0 Check 2m = 5 m= 5 2 5 or { 2.5} 2
- 11. 7. Solve x3 + 2x2 = 15x x3 + 2x2 - 15x = 0 Set = 0 x(x + 2x - 15) = 0 2 Factor Split/Solve x(x + 5)(x - 3) = 0 Check x = 0 or x + 5 = 0 or x - 3 = 0 {0, -5, 3}
- 12. Solve a2 – 3a = 40 1. {-8, 5} 2. {-5, 8} 3. {-8, -5} 4. {5, 8}
- 13. Solve 4r3 – 16r = 0 1. {-16, 4} 2. {-4, 16} 3. {0, 2} 4. {0, 4} 5. {-2, 0, 2} The degree will tell you how many answers you have!
- 14. Maria told this puzzle to her friends. “The product of four times my age and 45 less than three times my age is zero. How old am I?” Find Maria’s age. Let m = Maria’s age. 4m(3m - 45) = 0 4m = 0 or 3m - 45 = 0 m = 0 or 3m = 45 m = 0 or m = 15 0 is not reasonable so Maria is 15 years old!!
- 15. Find two consecutive integers whose product is 240. Let n = 1st integer. Let n + 1 = 2nd integer. Set = 0 n(n + 1) = 240 Factor Split/Solve n2 + n = 240 Check n2 + n – 240 = 0 (n – 15)(n + 16) = 0
- 16. (n – 15)(n + 16) = 0 n – 15 = 0 or n + 16 = 0 n = 15 or n = -16 The consecutive integers are 15, 16 or -16, -15.