Solving Quadratic Equations

Quadratic EquationsZero 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• second degree equation, also called quadratic equations

23 2 4x x

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 2nd degree equations.

subtract 2 on both sides divide both sides by 4

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

24 2 18x 2 2

24 16x 24 16

4 4

x

2 4x 2x

Solving a 2nd Degree Equation

• What about this one?

Or this one?

We need new strategy…

24 16x x

22 11 12x x

Lets go over some vocabulary

2nd 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 00 • 0 = 0

If A • B = 0We 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 = 0Solve each equation.

x = -3 or x = 5

Solve (x + 3)(x - 5) = 0

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}

Solve (y – 3)(2y + 6) = 0

1. {-3, 3}2. {-3, 6}3. {3, 6}4. {3, -6}

Solving a Quadratic Equation

• What about this one?

Or this one?

Lets take them one at a time

24 16x x

22 11 12x x

Solving a Quadratic Equation• What about this one? if we are going to use the zero product property, it needs to equal zero It also has to be the product of 2 factors We have to factor it Find GCF, then divide by it

24 16x x16x16x

24 16 0x x

4 (x

4x

4) 0x

Solving a Quadratic Equation• What about this one? if we are going to use the zero product property, it needs to equal zero Now break up the two factors and make each equal to zero Then solve each

24 16x x16x16x

24 16 0x x

4 (x

4x

4) 0x

4 0x 4and x 0x

4 0x

Solve x2 - 11x = 0

GCF = xx(x - 11) = 0

x = 0 or x - 11 = 0x = 0 or x = 11

{0, 11}

Solve a2 - 24a +144 = 0

a2 - 24a + 144 = 0(a - 12)(a - 12) = 0

a - 12 = 0a = 12{12}

a2

144

144a2

-24a

-12a -12a -12a

-12a

a -12a-12

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 order2. Squared term has to be

positive3. Put 1st and 3rd term in

the box4. We have two boxes and

1 term left, the xgame will show us how to split them up.

5. Multiply the 1st and 3rd 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 x2 + 2x = 15

x2 + 2x – 15 = 0 (x - 3)(x + 5) = 0 x – 3 = 0 x + 5 = 0 x = 3 or x = -5 {3, -5}

x2

-15

-15x2

2x

5x -3x 5x -3x

x +5x -3

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 order2. Squared term has to be

positive3. Put 1st and 3rd term in

the box4. We have two boxes and

1 term left, the xgame will show us how to split them up.

5. Multiply the 1st and 3rd 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

x2 + 2x – 15 = 0

1. Set the equation equal to 0.2. 2 terms, factor by distribution or or difference of two squares3. 4 terms, reverse box4. 3 terms, X boxNo matter how many terms, always start

by finding the GCF

4 steps for solving a quadratic equation by factoring