Solving Quadratic Equations with a leading coefficient other than 1 8-7 Notes for Algebra 1 Solving Quadratic Equations with a leading coefficient other than 1

8-7 pg. 513 10-30, 45-72(x3)

Factoring: 𝑎𝑥 2 +𝑏𝑥+𝑐 𝑎𝑥 2 +𝑏𝑥+𝑐 Find factors of 𝑎𝑐 that will add together to give you the middle term 𝑎𝑥 2 +𝑏𝑥−𝑐 Find factors of 𝑎𝑐 that will subtract from each other to give you the middle term.

Factoring using the Box Method 𝑥 2 +𝑏𝑥+𝑐 𝐹𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚×𝑙𝑎𝑠𝑡 𝑡𝑒𝑟𝑚 𝑥 𝑧 𝑐𝑥 2 = 𝑚 𝑝 List all factors of the product above, find the factors that will add them 𝑥 together to get the middle term 𝑏𝑥= 𝑚 + 𝑝 Place these factors in the other two 𝑤 squares and Pull the GCF up and left. 𝑥+𝑤 𝑥+𝑧 𝐹𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚 𝑚 𝑝 𝐿𝑎𝑠𝑡 𝑇𝑒𝑟𝑚

2𝑥 2 4𝑥 3𝑥 6 Example of Box method 2 𝑥 2 +7𝑥+6 12𝑥 2 Sum 𝑥 2 1𝑥∙12𝑥 13𝑥 −1𝑥∙−12𝑥 −13𝑥 2𝑥 2𝑥∙6𝑥 8𝑥 −2𝑥∙−6𝑥 −8𝑥 3𝑥∙4𝑥 7𝑥 3 −3𝑥∙−4𝑥 −7𝑥 𝑥+2 𝑥−3 2𝑥 2 4𝑥 3𝑥 6

2𝑥 2 −4𝑥 3𝑥 −6 Example of Box method 2 𝑥 2 −𝑥−6 −12𝑥 2 Sum 𝑥 −2 1𝑥∙−12𝑥 −11𝑥 −1𝑥∙12𝑥 11𝑥 2𝑥 2𝑥∙−6𝑥 −4𝑥 −2𝑥∙6𝑥 4𝑥 3𝑥∙−4𝑥 −𝑥 3 −3𝑥∙4𝑥 𝑥 𝑥−2 𝑥+3 2𝑥 2 −4𝑥 3𝑥 −6

Example 1: Factoring 𝑎𝑥 2 +𝑥+𝑐 Factor each Trinomial. 1.) 5𝑥 2 +27𝑥+10 2.) 4𝑥 2 +24𝑥+32

Example 1: Factoring 𝑎𝑥 2 +𝑥+𝑐 Factor each Trinomial. 1.) 5𝑥 2 +27𝑥+10 2.) 4𝑥 2 +24𝑥+32 5𝑥+2 𝑥+5 4 𝑥+2 𝑥+4

Example 2: Factor 𝑎𝑥 2 −𝑏𝑥+𝑐

Example 2: Factor 𝑎𝑥 2 −𝑏𝑥+𝑐

Prime Polynomial A polynomial that cannot be written as a product of two polynomials with integral coefficients.

Example 3: Determine whether a polynomial is Prime Factor. If the polynomial cannot be factored using integers, write prime. 3𝑥 2 +7𝑥−5

Example 3: Determine whether a polynomial is Prime Factor. If the polynomial cannot be factored using integers, write prime. 3𝑥 2 +7𝑥−5 Prime

Example 4: Solve Equations by Factoring MODEL ROCKETS Mr. Bird’s science class built a model rocket. They launched the rocket outside. It cleared the top of a 60-foot high pole and then landed in a nearby tree. If the launch pad was 2 feet above the ground, the initial velocity of the rocket was 64 feet per second, and the rocket landed 30 feet above the ground, how long was the rocket in flight? Use the equation ℎ=−16𝑡 2 +𝑣𝑡+ ℎ 0

Example 4: Solve Equations by Factoring MODEL ROCKETS Mr. Bird’s science class built a model rocket. They launched the rocket outside. It cleared the top of a 60-foot high pole and then landed in a nearby tree. If the launch pad was 2 feet above the ground, the initial velocity of the rocket was 64 feet per second, and the rocket landed 30 feet above the ground, how long was the rocket in flight? Use the equation ℎ=−16𝑡 2 +𝑣𝑡+ ℎ 0 ≈3.5 𝑠𝑒𝑐𝑜𝑛𝑑𝑠