5.2: Solving Quadratic Equations by Factoring

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I can use the zero product property to solve quadratics by factoring

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5.2: Solving Quadratic Equations by Factoring (p. 256) How do you factor: x2 +bx +c ax2+bx+c a2 −b2 a2 +2ab +b2 a2 −2ab +b2 How do you solve a quadratic function? Everything you ever learned about factoring in one section!

To solve a quadratic eqn To solve a quadratic eqn. by factoring, you must remember your factoring patterns!

Zero Product Property Let A and B be real numbers or algebraic expressions. If AB=0, then A=0 or B=0. This means that If the product of 2 factors is zero, then at least one of the 2 factors had to be zero itself!

Example: Solve. x2+3x-18=0 x2+3x-18=0 Factor the left side (x+6)(x-3)=0 set each factor =0 x+6=0 OR x-3=0 solve each eqn. -6 -6 +3 +3 x=-6 OR x=3 check your solutions!

Example: Solve. 2t2-17t+45=3t-5 2t2-17t+45=3t-5 Set eqn. =0 2t2-20t+50=0 factor out GCF of 2 2(t2-10t+25)=0 divide by 2 t2-10t+25=0 factor left side (t-5)2=0 set factors =0 t-5=0 solve for t +5 +5 t=5 check your solution!

Example: Solve. 3x-6=x2-10 3x-6=x2-10 Set = 0 0=x2-3x-4 Factor the right side 0=(x-4)(x+1) Set each factor =0 x-4=0 OR x+1=0 Solve each eqn. +4 +4 -1 -1 x=4 OR x=-1 Check your solutions!

Example: Factor 3x2 −17x+10 1. Factors of (3)(10) that add to −17 2. Factor by grouping 3. Rewrite equation 4. Use reverse distributive 5. Answer 3x2 −17x+10 2. 3x2 −?x −?x+10 3. 3x2 −15x −2x+10 4. 3x(x−5)−2(x−5) 5. (x−5)(3x−2)

Example: Factor 3x2 −17x+10 1.Rewrite the equation 2. 3x2 −?x −?x+10 3. 1.Rewrite the equation 2. Factors of (3)(10) that add to −17 (−15 & −2) 3. Place numbers in a box 4. Take our common factors in rows. 5. Take our common factors in columns. x −5 3x2 −15x 3x −2 −2x +10

Finding the Zeros of an Equation The Zeros of an equation are the x-intercepts ! First, change y to a zero. Now, solve for x. The solutions will be the zeros of the equation.

Example: Find the Zeros of y=x2-x-6 y=x2-x-6 Change y to 0 0=x2-x-6 Factor the right side 0=(x-3)(x+2) Set factors =0 x-3=0 OR x+2=0 Solve each equation +3 +3 -2 -2 x=3 OR x=-2 Check your solutions! If you were to graph the eqn., the graph would cross the x-axis at (-2,0) and (3,0).

Assignment p. 260, 23-31 odd 35-43 odd, 47-51 odd, 57-69 odd