Section 5.8 Solving Quadratic Equations by Factoring Chapter 5: Factoring Zero Product Property: • • Definition written using words. If the product of 2 or more factors equals zero, then at least one of the factors is equal to zero. Definition written using symbols: If 𝑎 × 𝑏 = 0 𝑡ℎ𝑒𝑛 𝑒𝑖𝑡ℎ𝑒𝑟 𝑎 = 0 𝑜𝑟 𝑏 = 0 𝑜𝑟 𝑏𝑜𝑡ℎ 𝑎 𝑎𝑛𝑑 𝑏 𝑒𝑞𝑢𝑎𝑙 0 Steps to solve an equation using the 𝑧𝑒𝑟𝑜 − 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 property: Step 1: Set the equation equal to zero Step 2: Factor the equation Step 3: Set each factor equal to zero and solve. (This is the step that uses the zero-product property) Step 4: Check NOTE: Factoring must be cleared if the problem is given in factored form and the problem is not set equal to zero. NOTE: This will be important when the factoring step involves a GCF. • • Common factors that have a variable usually give an answer of 0 Common factors that do NOT have a variable do not give a solution NOTE: There is a way to predict the number of solutions • • If the problem has no factoring, the highest power of the variable gives the maximum number of possible solutions. If the problem is completely factored, the number of factors that have variables gives the maximum number of possible solutions. Section 5.8 Solving Quadratic Equations by Factoring Chapter 5: Factoring For Example: Solve using the zero-product property: Step 1: step is already complete as the equation is set equal to zero. (𝑥 − 2)(𝑥 − 5) = 0 Step 2: step is already complete as the equation is factored. Step 1: Set the equation equal to zero Step 3: Step 2: Factor the equation Step 3: Set each factor equal to zero and solve. (This is the step that uses the zeroproduct property) 𝑥−2=0 +2 + 2 𝑥=2 𝑥−5=0 +5 + 5 𝑥=5 Answer: x = 2, 5 Step 4: Step 4: Check Check x = 2 Check x = 5 (2 − 2)(2 − 5) = 0 (0)(−3) = 0 0=0✓ (5 − 2)(5 − 5) = 0 (3)(0) = 0 0= 0✓ Section 5.8 Solving Quadratic Equations by Factoring Factoring #1 - 21: Solve each equation. 1) (𝑥 – 3)(𝑥 + 2) = 0 2) (𝑥 + 3)(𝑥 – 5) = 0 3) (𝑥 – 1)(𝑥 – 7) = 0 4) (𝑥 – 9)(𝑥 – 10) = 0 Chapter 5: 5) (3𝑥 + 12)(2𝑥 – 8) = 0 6) (2𝑥 – 12)(3𝑥 + 15) = 0 7) (2𝑥 – 9)(3𝑥 + 10) = 0 8) (5𝑥 + 4)(2𝑥 – 3) = 0 9) (5𝑥 – 7)(3𝑥 – 11) = 0 10) 3𝑥(𝑥 – 4)(𝑥 + 5) = 0 11) 7𝑥(𝑥 – 1)(𝑥 + 2) = 0 12) 4𝑥(2𝑥 – 10)(3𝑥 + 15) = 0 13) 5𝑥(2𝑥 + 10)(4𝑥 + 20) = 0 14) 𝑥(𝑥 + 1)(𝑥 + 2) = 0 15) 𝑥(𝑥 – 3)(𝑥 + 5) = 0 16) 𝑥(7𝑥 – 10)(3𝑥 – 16) = 0 17) 7(𝑥 – 1)(𝑥 – 2) = 0 18) 4(𝑥 – 3)(𝑥 + 5) = 0 19) 2(5𝑥 – 30)(3𝑥 + 18) = 0 20) 11(4𝑥 – 1)(3𝑥– 5) = 0 21) 6(2𝑥 – 9)(5𝑥 – 1) = 0 For Example: Solve using the zeroproduct property: Step 1: step is already complete as the equation is set equal to zero. 2𝑥 2 − 3𝑥 − 5 = 0 Step 2: Factor by grouping (see next page for factoring) 2𝑥 2 − 3𝑥 − 5 = 0 Step 1: Set the equation equal to zero (2𝑥 − 5)(𝑥 + 1) = 0 Step 2: Factor the equation Step 3: Set each factor equal to zero and solve. (This is the step that uses the zero-product property) Step 4: Check Steps for factoring are on the next page: Step 3: 2𝑥 − 5 = 0 +5 + 5 2𝑥 = 5 𝑥+1=0 −1 − 1 𝑥 = −1 5 𝑥=2 5 Answer: x = 2, −1 Step 4: 5 Check x = 2 5 5 (2 ( ) − 5) ( + 1) = 0 2 2 5 2 (5 − 5) (( + )) = 0 2 2 (0)(7⁄2) 0=0✓ Check x = -1 (2(−1) − 5)(−1 + 1) = 0 (−7)(0) = 0 0=0✓ Factor: 2𝑥 2 − 3𝑥 − 5 Step 1: There is no GCF, proceed to step 2. Step 2: Multiply “a” and “c” 𝑎𝑐 = (2)(−5) = −10 Find the +, −𝑓𝑎𝑐𝑡𝑜𝑟𝑠𝑜𝑓 −10, 𝑠𝑚𝑎𝑙𝑙𝑒𝑟 𝑛𝑢𝑚𝑏𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 −10 = 2(−5) −10 = 1(−10) Step 3: Find the pair that add to −3 −10 = 2(−5) 2 + (−5) = −3 −10 = 1(−10) 1 + (−10+ = −9 Step 4: Rewrite the −3𝑥 using the factors of 2 and −5 −3𝑥 = −5𝑥 + 2𝑥 (It is preferred to write the negative first.) Step 5: 2𝑥 2 − 3𝑥 − 5 = 2𝑥 2 − 5𝑥 + 2𝑥 − 5 Step 6: Factor 2𝑥 2 − 5𝑥 + 2𝑥 − 10 by grouping. 2𝑥 2 − 5𝑥 | + 2𝑥 − 5 GCF of 2x2 – 5x GCF = x 𝑥 2 − 5𝑥 = 𝑥(2𝑥 − 5) GCF of 2x - 10 GCF = 1 2𝑥 – 1 = 1(2𝑥 − 5) 𝑥 2 − 5𝑥 + 2𝑥 − 10 = 𝑥(2𝑥 − 5) + 1(2𝑥 − 5) = (2𝑥 − 5)(𝑥 + 1) is the answer. Step 7: Check (2𝑥 − 5)(𝑥 + 1) =2𝑥 2 + 2𝑥 − 5𝑥 − 5 = 5𝑥 2 − 3𝑥 − 5 ✓ Firsts 2𝑥𝑥 = 2𝑥 2 Outers 2𝑥(1) = 2𝑥 Inners -5𝑥 Lasts -5(1) = −5 #22 - 42: Solve each equation. 23) 𝑥 2 – 14𝑥 + 45 = 0 22) 𝑥 2 – 5𝑥 + 6 = 0 25) 𝑥 2 – 5𝑥 – 6 = 0 24) 𝑥 2 + 6𝑥 – 7 = 0 27) 𝑏 2 – 25 = 0 26) 𝑦 2 – 81 = 0 29) 49𝑦 2 – 16 = 0 28) 25𝑥 2 – 16 = 0 31) 5𝑥 2 + 9𝑥 + 4 = 0 30) 3𝑥 2 + 5𝑥 + 2 = 0 33) 3𝑥 2 – 5𝑥 – 2 = 0 32) 2𝑥 2 + 3𝑥 + 1 = 0 35) 2𝑥 2 + 5𝑥 + 3 = 0 34) 5𝑥 2 + 12𝑥 + 4 = 0 37) 𝑥 2 – 𝑥 – 6 = 0 36) 3𝑥 2 – 𝑥 – 2 = 0 39) 25𝑦 2 – 81 = 0 38) 𝑥 2 + 11𝑥 + 10 = 0 41) 3𝑦 2 + 5𝑦 – 2 = 0 40) 16𝑏 2 – 25 = 0 42) 6𝑥 2 – 𝑥 – 5 = 0 Section 5.8 Solving Quadratic Equations by Factoring Chapter 5: Factoring For Example: Solve using the zeroproduct property: Step 1: step is already complete as the equation is set equal to zero. 𝑥 3 − 6𝑥 2 + 8𝑥 = 0 Step 2: Factor (see next page for factoring) 𝑥 3 − 6𝑥 2 + 8𝑥 = 0 𝑥(𝑥 − 2)(𝑥 − 4) = 0 Step 1: Set the equation equal to zero Step 3: 𝑥=0 Step 2: Factor the equation 𝑥−2=0 +2 + 2 𝑥=2 Step 3: Set each factor equal to zero and solve. (This is the step that uses the zero-product property) S Answer: 𝑥 = 0,2,4 Step 4: tep 4: Check Check 𝑥 = 0 Steps for factoring are on the next page: (0)3 − 6(0)2 + 8(0) = 0 0−0+0= 0 0=0✓ Check 𝑥 = 2 (2)3 − 6(2)2 + 8(2) = 0 8 − 24 + 16 = 0 0=0✓ Check x = 4 (4)3 − 6(4)2 + 8(4) = 0 64 − 96 + 32 = 0 0=0✓ 𝑥−4=0 +4 + 4 𝑥=4 Factor: 𝑥 3 − 6𝑥 2 + 8𝑥 Step 1: There is a GCF of x. I need to factor out the GCF first. 𝑥 3 − 6𝑥 2 + 8𝑥 = 𝑥(𝑥 2 − 6𝑥 + 8) Steps 2 through Step 4 are used to factor what is left inside the parenthesis, 𝑥 2 − 6𝑥 + 8. I will keep the GCF of x in my answer, but I will not use the GCF in steps 2 through 4. Factor: 𝑥 2 − 6𝑥 + 8 Find the (-)(-) factors of 8 8 = (−1)(−8) 8 = (−2)(−4) Step 3: 8 = (−1)(−8) −1 + (−8) = −9 8 = (−2)(−4) −2 + (−4) = −6 Step 4: 𝑥 2 − 6𝑥 + 8 = (𝑥 − 2)(𝑥 − 4) (this is not my answer, as it is missing the GCF and it does not multiply to equal the original problem.) Add the GCF to complete the factoring. ANSWER: 𝑥 3 − 6𝑥 2 + 8𝑥 = 𝑥(𝑥 − 2)(𝑥 − 4) Step 5: Check: 𝑥(𝑥 − 2)(𝑥 − 4) rewrite so FOIL can be performed first = (𝑥 − 2)(𝑥 − 4)𝑥 =(𝑥 2 − 4𝑥 − 2𝑥 + 8)𝑥 = (𝑥 2 − 6𝑥 + 8)𝑥 =𝑥 2 (𝑥) + (−6𝑥)𝑥 + 8(𝑥) = 𝑥 3 + (−6𝑥 2 ) + 8𝑥 = 𝑥 3 − 6𝑥 2 + 8𝑥 ✓ #43 - 60: Solve each equation. (Remember to factor out the GCF first) 43) 3𝑥 3 + 5𝑥 2 + 2𝑥 = 0 44) 10𝑥 3 + 18𝑥 2 + 8𝑥 = 0 45) 4𝑥 2 + 2𝑥 – 6 = 0 46) 3𝑥 2 + 18𝑥 – 21 = 0 47) 5𝑥 2 – 20 = 0 48) 2𝑥 2 – 18 = 0 49) 3𝑥 3 – 15𝑥 2 + 18𝑥 = 0 50) 2𝑥 3 – 28𝑥 2 + 90𝑥 = 0 51) 3𝑥 2 – 15𝑥 – 18 = 0 52) 2𝑥 2 – 28𝑥 + 90 = 0 53) 10𝑥 2 + 18𝑥 + 8 = 0 54) 14𝑥 2 + 35𝑥 + 21 = 0 55) 10𝑥 3 + 18𝑥 2 + 8𝑥 = 0 56) 14𝑥 3 + 35𝑥 2 + 21𝑥 = 0 57) −2𝑥 2 + 10𝑥 + 12 = 0 58) −3𝑥 2 – 18𝑥 + 21 = 0 59) 𝑥 3 – 49𝑥 = 0 60) 𝑦 3 – 25𝑦 = 0 Section 5.8 Solving Quadratic Equations by Factoring For Example: Solve using the zeroproduct property: 2𝑥(𝑥 + 1) = 40 Step 1: Set the equation equal to zero Step 2: Factor the equation Step 3: Set each factor equal to zero and solve. (This is the step that uses the zero-product property) Step 4: Check Step 4: Check 𝑥 = 2 2(2)(2 + 1) = 40 (2)(2)(3) = 40 12 = 40 𝑡ℎ𝑖𝑠 𝑐𝑜𝑛𝑓𝑖𝑟𝑚𝑠 2 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑠𝑜𝑙𝑢𝑖𝑜𝑛 Check 𝑥 = −5 2(−5)(−5 + 1) = 40 (2)(−5)(−4) = 40 40 = 40 ✓ Check x = 4 2(4)(4 + 1) = 40 (2)(4)(5) = 40 40 = 40 ✓ Chapter 5: Factoring Step 1: I need to set the equation equal to zero and clear the factoring. 2𝑥(𝑥 + 1) = 40 2𝑥 2 + 2𝑥 = 40 2𝑥 2 + 2𝑥 − 40 = 0 Step 2: Factor 2𝑥 2 + 2𝑥 − 40 = 0 2(𝑥 2 + 𝑥 − 20) = 0 2(𝑥 + 5)(𝑥 − 4) = 0 Step 3: 2=0 No solution when GCF is a constant 𝑥+5=0 𝑥 = −5 Answer: 𝑥 = −5,4 𝑥−4=0 𝑥=4 #61 - 78: Solve each equation. 61) 𝑟(𝑟 + 1) = 12 62) 𝑥(𝑥 – 1) = 6 63) 𝑥(𝑥 – 4) = −3 64) 𝑦(𝑦 – 5) = −4 65) (𝑥 – 2)(𝑥 – 3) = 6 66) (𝑥 – 1)(𝑥– 2) = 12 67) (𝑥 + 1)(𝑥 – 4) = 6 68) (𝑧 + 3)(𝑧 – 4) = 8 69) 3𝑥(𝑥 + 1) = 6 70) 4𝑥(𝑥 – 4) = −12 71) (2𝑥 – 3)(𝑥 + 2) = 4 72) (3𝑥 – 4)(𝑥 – 5) = −6 73) 𝑥(𝑥 – 3) + 2 = 30 74) 3𝑥(𝑥 – 1) – 4 = 2 75) 2𝑥(𝑥 – 3) = 5𝑥(𝑥– 4) + 8 76) 2(3 – 4𝑥) = 𝑥(𝑥 – 1) – 2 77) 𝑥(𝑥 – 4) + 1 = 𝑥 + 7 78) 𝑥(𝑥 – 3) + 4 = 3𝑥 – 4