chapter 1.4 quadratic equations. quadratic equation in one variable an equation that can be written...
TRANSCRIPT
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