Section 8.1 Completing the Square. Factoring Before today the only way we had for solving quadratics...
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Transcript of Section 8.1 Completing the Square. Factoring Before today the only way we had for solving quadratics...
FactoringFactoring
• Before today the only way we had for solving quadratics was to factor.
x2 - 2x - 15 = 0
(x + 3)(x - 5) = 0
x + 3 = 0 or x - 5 = 0
x = -3 or x = 5
x = {-3, 5}
Zero-factorproperty
FactoringFactoring
x2 = 9
x2 - 9 = 0
(x + 3)(x - 3) = 0
x + 3 = 0 or x - 3 = 0
x = -3 or x = 3
x = {-3, 3}
Zero-factorproperty
Square Root PropertySquare Root Property
• If x and b are complex numbers and if x2 = b, then
bx OR bx
Solve each equation. Write radicals in simplified form.
112 b
11b Square Root Property
Radical will not simplify.
Solve each equation. Write radicals in simplified form.
122 c
12c
32c
Solution Set
Square Root Property
Solve each equation. Write radicals in simplified form.
36)2( 2 m
36)2( m
62 m
62 m22 8m62 m22
4m
}4,8{m
Solving Quadratic Equations by Solving Quadratic Equations by Completing the SquareCompleting the Square
x2 - 2x - 15 = 0
(x + 3)(x - 5) = 0
x + 3 = 0
or x - 5 = 0
x = -3 or x = 5
x = {-3, 5}
01522 xx
1522 xx
Now take 1/2 of the coefficient of x.
Square it.Add the result to both sides.
11Factor the left.
Simplify the right.16)1( 2 x161 x41x}5,3{x
Square Root Property
Completing the Square1. Divide by the coefficient
of the squared term.
2. Move all variables to one side and constants to the other.
3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.
4. Factor the left hand side and simplify the right.
5. Root and solve.
2 2 24 0x x 1 1 1 1
2 2 24 0x x 2 2 24x x
2
12 1
2
1 1
1 2
1 25x 1
21 25x
Completing the Square1. Divide by the coefficient
of the squared term.
2. Move all variables to one side and constants to the other.
3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.
4. Factor the left hand side and simplify the right.
5. Root and solve.
21 25x
1 5x
1 5x
{ 4,6}x
Completing the Square1. Make the coefficient of
the squared term =1.
2. Move all variables to one side and constants to the other.
3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.
4. Factor the left hand side and simplify the right.
5. Root and solve.
0253 2 xx3333
03
2
3
52 xx
3
2
3
52 xx1 5 5
2 3 6
2 25 5
6 6
36
49
6
52
x
Completing the Square1. Make the coefficient of the squared term =1.
2. Move all variables to one side and constants to the other.
3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.
4. Factor the left hand side and simplify the right.
5. Root and solve.
25 49
6 36x
36
49
6
5x
6
7
6
5x
26
12
6
7
6
5
x
3
1
6
2
6
7
6
5x
3
1,2x
1. Make the coefficient of the squared term =1.
2. Move all variables to one side and constants to the other.
3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.
4. Factor the left hand side and simplify the right.
5. Root and solve.
2 5 3 0x x 2 5 3x x
1 5 5
2 1 2
252
25 37
2 4x
252
5 37
2 2x
5 37
2x
Deriving The Quadratic Formula
2 0b c
x xa a
Divide both sides by a
2 22
2 2
b b c bx x
a a a a
2 2
2 2
4
2 4 4
b b acx
a a a
2
2
4
2 4
b b acx
a a
2 4
2
b b acx
a
Complete the square by adding (b/2a)2 to both sides
Factor (left) and find LCD (right)
Combine fractions and take the square root of both sides
Subtract b/2a and simplify
2If 0 (and 0 then:),ax bx c a