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Chapter 1.4 Quadratic Equations
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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 Equation
Factoring 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 = k
x2 – 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 = 1
Solve x2 – 4x – 14 = 0
Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0
144xx2
Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0
144xx2 2
4
Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0
144xx2 2
42
Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0
144xx2 2
42
22
Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0
144xx2 2
42
22 4
Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0
144xx2 2
42
22 441444xx2
Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0
144xx2 2
42
22 441444xx2 81
Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0
144xx2 2
42
22 441444xx2 81
2)2( x
Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0
144xx2 2
42
22 441444xx2 81
18)2( 2 x
Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0
144xx2 2
42
22 441444xx2 81
18)2( 2 x
182x
Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0
144xx2 2
42
22 441444xx2 81
18)2( 2 x
182x 29
Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0
144xx2 2
42
22 441444xx2 81
18)2( 2 x
182x 29 23
Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0
144xx2 2
42
22 441444xx2 81
18)2( 2 x
182x 29 23
232x
Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0
144xx2 2
42
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
09
1x
9
12x2
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
09
1x
9
12x2
09
1x
3
4x2
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
09
1x
9
12x2
9
1 x
3
4x2
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
09
1x
9
12x2
9
1 x
3
4x2
2
3
4
2
1
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
09
1x
9
12x2
9
1 x
3
4x2
2
3
4
2
1
2
3
2
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
09
1x
9
12x2
9
1 x
3
4x2
2
3
4
2
1
2
3
2
9
4
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
09
1x
9
12x2
9
1 x
3
4x2
2
3
4
2
1
2
3
2
9
4
9
4
9
1
9
4x
3
4x2
Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0
09
1x
9
12x2
9
1 x
3
4x2
2
3
4
2
1
2
3
2
9
4
9
4
9
1
9
4x
3
4x2
9
5
Example 4 Using the Method of Completing the Square, a ≠1
9
5
9
4x
3
4x2
Example 4 Using the Method of Completing the Square, a ≠1
9
5
9
4x
3
4x2
9
5
3
2-x
2
Example 4 Using the Method of Completing the Square, a ≠1
9
5
9
4x
3
4x2
9
5
3
2-x
2
9
5
3
2-x
Example 4 Using the Method of Completing the Square, a ≠1
9
5
9
4x
3
4x2
9
5
3
2-x
2
9
5
3
2-x
9
5
3
2x
Example 4 Using the Method of Completing the Square, a ≠1
9
5
9
4x
3
4x2
9
5
3
2-x
2
9
5
3
2-x
9
5
3
2x
9
5
3
2x
Example 4 Using the Method of Completing the Square, a ≠1
9
5
9
4x
3
4x2
9
5
3
2-x
2
9
5
3
2-x
9
5
3
2x
9
5
3
2x
3
5
3
2x
Example 4 Using the Method of Completing the Square, a ≠1
9
5
9
4x
3
4x2
9
5
3
2-x
2
9
5
3
2-x
9
5
3
2x
9
5
3
2x
3
5
3
2x
3
52x
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 Equation
Solve x3 + 8 = 0
Example 8 Solving a Variable That is Squared
Solve for the specified variable.
dd
A for ,4
2
Example 8 Solving a Variable That is Squared
Solve 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.
a
acbbx
2
42 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 Discriminant
Determine the number of solutions and tell whether they are rational, irrational, or nonreal complex numbers.
5x2 + 2x – 4 = 0a
acbbx
2
42
) (2
) )( (4) () ( 2 x
a
bc
Example 9 Using the Discriminant
Determine the number of solutions and tell whether they are rational, irrational, or nonreal complex numbers.
x2 – 10x = -25a
acbbx
2
42
) (2
) )( (4) () ( 2 x
a
bc
Example 9 Using the Discriminant
Determine the number of solutions and tell whether they are rational, irrational, or nonreal complex numbers.
2x2 – x + 1 = 0a
acbbx
2
42
) (2
) )( (4) () ( 2 x
a
bc
Homework 1.4 # 1-79
