Solving Quadratic Equations Quadratic Equations Zero Product Property Using Factoring to solve Quadratic Equations

Degree of an equation • Equations with one variable, the degree is equal to the highest exponent • • 3x + 4 = 13 first degree equation 3

x

2  2

x

 4 second degree equation, also called quadratic equations

Solving a First Degree Equation • You have had a good amount of experience with this: 3x + 4 = 13 to solve, get the variable alone - 4 -4 subtract 4 on both sides 3x = 9 divide both sides by 3 x = 3 variable alone, coefficient of 1 3(3) + 4 = 13 3 is the only number that makes 9 + 4 = 13 this equation true

Solving a Second Degree Equation • Getting the variable alone, with a coefficient of one will work in some 2 nd degree equations.

4

x

2  2  18 subtract 2 on both sides  2 divide both sides by 4 4 4

x

2

x

 4   2 take the square root of both sides 2 or -2 can make the equation true

Solving a 2 nd Degree Equation • What about this one?

4

x

2   16

x

Or this one?

2

x

2  11

x

  12 We need new strategy…

Lets go over some vocabulary

2 nd

degree equations—we are going to call them quadratic equations or quadratics

Lets go over some vocabulary

Factoring

a number or expression— means to break it down into two or more parts that are multiplied together.

Zero Product Property

If A • 5 = 0 then A = 0

Zero Product Property

If 5 • B = 0 then B = 0

Zero Product Property

If A • B = 0 then A = 0 or B = 0, 0 • B = 0 or A • 0 = 0 or both A and B equal 0 0 • 0 = 0

Solve (x + 3)(x - 5) = 0 If A • B = 0 We can use the Zero Product Property whenever we have two factors that equal zero we know that either A = 0 or B = 0

x + 3 = 0

or

x - 5 = 0

Solve each equation.

x = -3 or x = 5

Solve (2a + 4)(a + 7) = 0 A • B = 0 So….

2a + 4 = 0

or

a + 7 = 0

2a = -4 or a = -7 a = -2 or

{-2, -7}

Solve t(t - 3) = 0 A(B) = 0 So……..

t = 0

or

t - 3 = 0

or t = 3

{0, 3}

1. {-3, 3} 2. {-3, 6} 3. {3, 6} 4. {3, -6} Solve (y – 3)(2y + 6) = 0

Solving a Quadratic Equation • What about this one?

4

x

2   16

x

Or this one?

2

x

2  11

x

  12 Lets take them one at a time

Solving a Quadratic Equation • What about this one? 4

x

2  16

x

  16

x

 16

x

if we are going to use the zero product property, it needs to equal zero 4

x

2  16

x

 0 4

x x

0 It also has to be the product of 2 factors We have to factor it Find GCF, then divide by it

Solving a Quadratic Equation 4 • What about this one? 4

x

2  16

x

  16

x

 16

x

if we are going to use the zero product property, it needs to equal zero 4

x

2  16

x

 0 4

x x x

 0  0

x x

0   4 0 Now break up the two factors and make each equal to zero Then solve each

Solve x 2 - 11x = 0 GCF = x x(x - 11) = 0

x = 0

or

x - 11 = 0

x = 0 or x = 11

{0, 11}

Solve a a -12 2 - 24a +144 = 0 a a 2 -12a -12 -12a 144 a 2 -12a 144a 2 -12a -24a This one does not have a GCF other than 1. We use the X box to factor and solve this one. - 24a + 144 = 0 (a - 12)(a - 12) = 0 a - 12 = 0 a = 12

{12}

1. Put in descending order 2. Squared term has to be positive 3. Put 1 st and 3 rd term in the box 4. We have two boxes and 1 term left, the xgame will show us how to split them up. 5. Multiply the 1 st and 3 rd term and put on top of x 6. The middle term goes on the bottom 7. The numbers you find go in the two remaining boxes 8. Find the GCF of each column and row for your factors

Solve x x -3 x 2 x 2 x 2 -3x x 2 + 2x = 15 + 2x – 15 = 0 +5 5x -15 5x -15x 2 -3x + 2x – 15 = 0 (x - 3)(x + 5) = 0 x – 3 = 0 x + 5 = 0 x = 3 or x = -5

{3, -5}

2x This one does not have a GCF other than 1. We use the X box to factor and solve this one. 1. Put in descending order 2. Squared term has to be positive 3. Put 1 st and 3 rd term in the box 4. We have two boxes and 1 term left, the xgame will show us how to split them up. 5. Multiply the 1 st and 3 rd term and put on top of x 6. The middle term goes on the bottom 7. The numbers you find go in the two remaining boxes 8. Find the GCF of each column and row for your factors

4 steps for solving a quadratic equation by factoring

1. Set the equation equal to 0.

2. 2 terms, factor by distribution or or difference of two squares 3. 4 terms, reverse box 4. 3 terms, X box No matter how many terms, always start by finding the GCF