Solving Quadratic

Equations

x y-2-1134

Quadratic Equation

y = ax2 + bx + c

We will learn how to solve a quadratic equation by…

Taking the square rootFactoring and then solvingExamining a graph or tableUsing the Quadratic Formula

Square Root Method

When the equation is ax2 + c = 0: Isolate the x2, and take the square

root of both sides to cancel the exponent

Example:

6

6

6

244

0244

2

2

2

2

x

x

x

x

x Key points:Use ± to show that x was either positive or negative before being squared. Simplify the square root when necessary.

Factor and Solve, when a=1

From Math 1, you factored trinomials:x2 + 4x – 12 (x + 6)(x – 2)

When solving a similar trinomial equation, set each ( ) equal to 0 to solve for x:x2 + 4x – 12 = 0(x + 6)(x – 2) = 0x + 6 = 0 or x – 2 = 0

x = -6 or x = 2

Factor and Solve, when a > 1

In Math 2 you learned how to factor if there is a coefficient of the squared term:2x2 – 7x – 4 (2x + 1)(x – 4)

Solve an equation by setting each ( ) equal to 0 again, but expect some fractions!2x2 – 7x – 4 = 0(2x + 1)(x – 4) = 02x + 1 = 0 or x – 4 = 0x = -1/2 or x = 4

Since the number of solutions is the same as the number of x-intercepts, the number

of real solutions is at most two.

Solving Quadratic Equations with

Graphs and Tables

No solutions

6

4

2

-2

5

f x = x2-2 x +56

4

2

-2

5

2

-2

-4

-5 5

One solution Two solutions

Solving Equations

When we talk about solving these equations, we want to find the value of x when y = 0. These values, where the graph crosses the x-axis, are called the x-intercepts.

These values are also referred to as solutions, zeros, or roots.

Solutions to Quadratic Equations

These terms all mean the same thing.

solutions

x-intercep

tsroots

zeros

Example f(x) = x2 - 4

Identifying Solutions4

2

-2

-4

-5 5

Solutions are -2 and 2.

Now you try this problem.

What are the solutions to f(x) = 2x - x2 ?

Solutions are 0 and 2.

Identifying Solutions

4

2

-2

-4

5

One method of graphing uses a table with

x-values.Look at the graph of y = x2 - 4x

Roots 0 and 4

Look at the table of values that goes with the graph.

What do you noticeabout the y value for the roots?

Identifying Solutions from Tables

x y0 01 -32 -43 -34 0

4

2

-2

-4

5

0

0

Look at the parabola y = x2 - 2x - 8.

What are the solutions?

What would the table look like at those values?

Graphing Quadratic Equations

x y-2 0

-1 -5

1 -9

3 -5

4 0

2

-2

-4

-6

-8

5

0

0

The Quadratic Formula (from Math 2)

Yes! The Quadratic Formula is provided on your GHSGT formula sheet!

Where a, b, and c are the numbers (coefficients) from the equation:

ax2 + bx + c = 0

a

acbbx

2

42

Example

3x2 – 8x + 4 = 0

a

acbbx

2

42

6

4

6

126

48

6

168

6

48648

)3(2

)4)(3(4)8()8( 2

orx

x

x

x

x = 2 or x = 2/3

Complex Numbers

Recall from Math 2 that i=√-1, so that anytime you have a negative under the square root, simplify with i.

Example: √-64 = 8i Example: √-120 Factor Tree!

= 2i√30

Complex Solutions

Solve using Quadratic Formula:x2 – 5x + 9 = 0

2

115

2

115

2

36255

)1(2

)9)(1(4)5()5( 2

ix

x

x

a

acbbx

2

42