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