5.2: Solving Quadratic Equations by Factoring

(p. 256)How do you factor:

x2 +bx +cax2+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. 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. =02t2-20t+50=0 factor out GCF of 22(t2-10t+25)=0 divide by 2t2-10t+25=0 factor left side(t-5)2=0 set factors =0t-5=0 solve for t+5 +5t=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. 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)

1. Factors of (3)(10) that add to −17

2. Factor by grouping

3. Rewrite equation

4. Use reverse distributive

5. Answer

Example: Factor 3x2 −17x+10

1. 3x2 −17x+10

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 box4. Take our common factors in rows.5. Take our common factors in columns.

3x2 −15x

−2x +10

3x

−2

x −5

Finding the Zeros of an EquationFinding 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 ofy=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