Post on 11-Jan-2016
Solving Quadratic EquationsSection 1.3
What is a Quadratic Equation?
A quadratic equation in x is an equation that can be written in the standard form:
ax² + bx + c = 0Where a,b,and c are real numbers and
a ≠ 0.
Solving a Quadratic Equation by Factoring.
The factoring method applies the zero product property:
Words: If a product is zero, then at least one of its
factors has to be zero.
Math: If (B)(C)=0, then B=0 or C=0 or both.
Recap of steps for how to solve by Factoring
Set equal to 0 Factor Set each factor equal to 0 (keep the squared
term positive) Solve each equation (be careful when
determining solutions, some may be imaginary numbers)
Example 1Solve x² - 12x + 35 = 0 by factoring.
Factor:
Set each factor equal to zero by the zero product property.
Solve each equation to find solutions.
The solution set is:
(x – 7)(x - 5) = 0
(x – 7)=0(x – 5)=0
x = 7 or x = 5
{ 5, 7 }
Example 2Solve 3t² + 10t + 6 = -2 by factoring.
Check equation to make sure it is in standard form before solving. Is it?
It is not, so set equation equal to zero first:3t² + 10t + 8 = 0
Now factor and solve.(3t + 4)(t + 2) = 0
3t + 4 = 0 t +2 = 0
t = t = -23
4
Solve by factoring.
032 2 xx 032 xx
0x 032 x
2
3x
Solve by the Square Root Method.
If the quadratic has the form ax² + c = 0, where a ≠ 0, then we could use the square root method to solve.
Words: If an expression squared is equal to a constant, then that expression is equal to the positive or negative square root of the constant.
Math: If x² = c, then x = ±c.
Note: The variable squared must be isolated first (coefficient equal to 1).
Example 1:Solve by the Square Root Method:
2x² - 32 = 0
2x² = 32
x² = 16
=
x = ± 4
2x 16
Example 2:Solve by the Square Root Method.
5x² + 10 = 0
5x² = -10
x² = -2
x = ±
x = ±i 22
Example 3:Solve by the Square Root Method.
(x – 3)² = 25
x – 3 = ± 5
x – 3 = 5 or x – 3 = -5
x = 8 x = -2
Solve by the Square Root Method
1283 2 x
1283 x
3283 x 3283 x
3283 x3283 x
3
328x
3
328 x
Solve by Completing the Square.
Words Express the quadratic
equation in the following form.
Divide b by2 and square the result, then add the square to both sides.
Write the left side of the equation as a perfect square.
Solve by using the square root method.
Math x² + bx = c
x² + bx + ( )² = c + ( )²
(x + )² = c + ( )²
2
b
2
b
2
b
2
b
Example 1:Solve by Completing the Square.
x² + 8x – 3 = 0x² + 8x = 3
x² + 8x + (4)² = 3 + (4)² x² + 8x + 16 = 3 + 16
(x + 4)² = 19x + 4 = ±x = -4 ±
Add three to both sides.
Add ( )² which is (4)² to both sides.
Write the left side as a perfect square and simplify the right side.
Apply the square root method to solve.
Subtract 4 from both sides to get your two solutions.
2
b
19
19
Example 2:Solve by Completing the Square when the Leading Coefficient is not equal to 1.
2x² - 4x + 3 = 0
x² - 2x + = 0
x² - 2x + ___ = + ____
x² - 2x + 1 = + 1
(x – 1)² =
x – 1 = ±
x = 1 ±
Divide by the leading coefficient.
Continue to solve using the completing the square method.
Simplify radical.
2
3
2
3
2
3
2
5
2
5
2
10
Quadratic Formula
If ax² + bx + c = 0, then the solution is:
a
acbbx
2
42
If a quadratic can’t be factored, you must use the quadratic formula.
Solve
a
acbbx
2
42
12
11444 2
x
2
4164 x
0142 xx
a = 1
b = -4
c = -1
2
204 x
2
524 x
52 x
Solve 964 2 nn
0964 2 nn
42
94466 2
n
a
acbbn
2
42
8
144366 n
8
1806 n
8
566 n
4
533n
Solve xx 482
a
acbbx
2
42
0842 xx
12
81444 2
x
2
32164 x
2
164 x
2
44 ix
ix 22
Discriminant
The term inside the radical b² - 4ac is called the discriminant.
The discriminant gives important information about the corresponding solutions or answers of ax² + bx + c = 0, where a,b, and c are real numbers.
b² - 4ac Solutions
b² - 4ac > 0
b² - 4ac = 0
b² - 4ac < 0
a
acbbx
2
42
Tell what kind of solution to expect
0198282 xx
19814284 22 acb
792784
8