chapter 1.4
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Chapter 1.4. Quadratic Equations. Quadratic Equation in One Variable. An equation that can be written in the form ax 2 + bx + c = 0 where a, b, and c, are real numbers, is a quadratic equation. - PowerPoint PPT PresentationTRANSCRIPT
Chapter 1.4
Quadratic Equations
Quadratic Equation in One Variable
An equation that can be written in the form
ax2 + bx + c = 0
where a, b, and c, are real numbers, is a quadratic equation
A quadratic equation is a second-degree equation—that is, an equation with a squared term and no terms of greater degree.
x2 =25, 4x2 + 4x – 5 = 0,3x2 = 4x - 8
A quadratic equation written in the form
ax2 + bx + c = 0 is in standard form.
Solving a Quadratic EquationFactoring is the simplest method of solving a quadratic equation (but one not always easily applied).
This method depends on the zero-factor property.
Zero-Factor Property
If two numbers have a product of 0 then at least one of the numbers must be zero
If ab= 0 then a = 0 or b = 0
Example 1. Using the zero factor property.
Solve 6x2 + 7x = 3
A quadratic equation of the form x2 = k can also be solved by factoring.
x2 = kx2 – k=0
0 kxkx
0 kx 0or kx
kx kx or
property.root square theproves This
Square root property
If x2 = k, then
kx kx or
Example 2 Using the Square Root Property
Solve each quadratic equation.x2 = 17
Example 2 Using the Square Root Property
Solve each quadratic equation.x2 = -25
Example 2 Using the Square Root Property
Solve each quadratic equation.(x-4)2 = 12
Completing the Square
Any quadratic equation can be solved by the method of completing the square.
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
144xx2
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
144xx2 24
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
144xx2 24
2
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
144xx2 24
2
22
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
144xx2 24
2
22 4
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
144xx2 24
2
22 441444xx2
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
144xx2 24
2
22 441444xx2 81
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
144xx2 24
2
22 441444xx2 81
2)2( x
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
144xx2 24
2
22 441444xx2 81
18)2( 2 x
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
144xx2 24
2
22 441444xx2 81
18)2( 2 x
182x
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
144xx2 24
2
22 441444xx2 81
18)2( 2 x
182x 29
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
144xx2 24
2
22 441444xx2 81
18)2( 2 x
182x 29 23
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
144xx2 24
2
22 441444xx2 81
18)2( 2 x
182x 29 23
232x
Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0
144xx2 24
2
22 441444xx2 81
18)2( 2 x
182x 29 23
232x
232x
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
091x
912x2
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
091x
912x2
091x
34x2
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
091x
912x2
91 x
34x2
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
091x
912x2
91 x
34x2
2
34
21
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
091x
912x2
91 x
34x2
2
34
21
2
32
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
091x
912x2
91 x
34x2
2
34
21
2
32
94
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
091x
912x2
91 x
34x2
2
34
21
2
32
94
94
91
94x
34x2
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
091x
912x2
91 x
34x2
2
34
21
2
32
94
94
91
94x
34x2
95
Example 4 Using the Method of Completing the Square, a ≠1
95
94x
34x2
Example 4 Using the Method of Completing the Square, a ≠1
95
94x
34x2
95
32-x
2
Example 4 Using the Method of Completing the Square, a ≠1
95
94x
34x2
95
32-x
2
95
32-x
Example 4 Using the Method of Completing the Square, a ≠1
95
94x
34x2
95
32-x
2
95
32-x
95
32x
Example 4 Using the Method of Completing the Square, a ≠1
95
94x
34x2
95
32-x
2
95
32-x
95
32x
95
32x
Example 4 Using the Method of Completing the Square, a ≠1
95
94x
34x2
95
32-x
2
95
32-x
95
32x
95
32x
35
32x
Example 4 Using the Method of Completing the Square, a ≠1
95
94x
34x2
95
32-x
2
95
32-x
95
32x
95
32x
35
32x
352x
The Quadratic Formula
02 cbxax
aacbbx 2
42
Watch the derivation
Example 5 Using the Quadratic Formula(Real Solutions)Solve x2 -4x = -2
Example 6 Using the Quadratic Formula(Non-real Complex Solutions)Solve 2x2 = x – 4
Example 7 Solving a Cubic EquationSolve x3 + 8 = 0
Example 8 Solving a Variable That is SquaredSolve for the specified variable.
ddA for ,4 2
Example 8 Solving a Variable That is SquaredSolve for the specified variable.
trkstrt for ),0(2
The Discriminant The quantity under the radical in the quadratic formula, b2 -4ac, is called the discriminant.
aacbbx
242
Discriminant
Then the numbers a, b, and c are integers, the value of the discriminant can be used to determine whether the solution of a quadratic equation are rational, irrational, or nonreal complex numbers, as shown in the following table.
Discriminant Number of Solutions Kind of Solutions
Positive (Perfect Square)
Positive (but not a Perfect Square)
Zero
Negative
Two
Two
One (a double solution)
Two
Rational
Irrational
Rational
Nonreal complex
Example 9 Using the DiscriminantDetermine the number of solutions and tell whether they are rational, irrational, or nonreal complex numbers. 5x2 + 2x – 4 = 0
aacbbx
242
) (2) )( (4) () ( 2
x
a
bc
Example 9 Using the DiscriminantDetermine the number of solutions and tell whether they are rational, irrational, or nonreal complex numbers. x2 – 10x = -25
aacbbx
242
) (2) )( (4) () ( 2
x
a
bc
Example 9 Using the DiscriminantDetermine the number of solutions and tell whether they are rational, irrational, or nonreal complex numbers. 2x2 – x + 1 = 0
aacbbx
242
) (2) )( (4) () ( 2
x
a
bc
Homework 1.4 # 1-79