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

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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 12a12
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