2446

7221

4175

Practice 1 Practice 2

Practice 3

Warm Up Use a calculator to evaluate. Round the results to the nearest tenth.

2859

Practice 4

2446

7221

4175

3.6 3.0

Practice 1 Practice 2

5.0 3.0

Practice 3

2.0 3.2

Warm Up Use a calculator to evaluate. Round the results to the nearest tenth.

2859

6.11 6.2

Practice 4

9.4B Using the Discriminant

Algebra 1 Glencoe McGraw-Hill Linda Stamper

The x-intercepts of the graph are the solutions, or roots, of the related equation

cbxaxy 2

Because at the x-intercept, y is equal to 0.

0.cbxax2

cbxaxy 2

0cbxax2

function

related equation

Notice the first equation is equal to y and the second equation is equal to 0. Why is y = 0?

x

y

The solutions, or roots, are –2 and 2.

x

y

The solutions, or roots, are –4 and 1.

• • • •

The solutions, or roots, are NOT ordered pairs.

Write the quadratic formula:

2a

4acbbx

2

What happens to the quadratic equation if the value under the radical is a zero?

2a

bx

What is this the formula for?

If the vertex of the parabola is on the x-axis, there is one solution. It is the x-coordinate of the vertex.

x

y

There is one solution.

•

2a

4acbbx

2

What do you predict will happen when there is a negative under the radical?

What does the graph look like to support your answer?

If the parabola does not intersect the x-axis, then there is no real solution.

x

y

There are no real solutions, or roots.

2a4acbb

x2

b2 – 4ac

Another way to identify how many solutions there are to an equation, if any, is to use the discriminant. In the quadratic formula, the expression inside the radical is called the discriminant.

The Number of Solutions of a Quadratic EquationIf the discriminant is positive, then the equation has

two solutions.If the discriminant is zero, then the equation has one solution.If the discriminant is negative, then the equation has no real solution.

discriminant

State the value of the discriminant. Then determine the number of real roots of the equation.Write problem. 015x3x2 Identify values for a, b and c.

1c ,5b ,3a

Write discriminant expression.

4acb2 Substitute values for a, b, and c. (Place values in parentheses to avoid sign errors and change subtraction to addition.)

1345 2

11225

1225 13

Simplify to find the discriminant.Identify the number of solutions.

two

4acb2 5244 2

State the value of the discriminant. Then determine the number of real roots of the equation.

2x42x Example 1

Example 2

Example 302510xx2

054x2x2

4acb2 4142 2

no real roots

04x2x2

444

4acb2 2514102

254100

100100 0

one

5816

4016 56

x452x2

two

164 12

36xxy 2

4acb2 3146 2

two

036xx2

3436 What is

the related equatio

n?

Example 4

1236 24

Example 5

Example 6106xxy 2 96xxy 2

4acb2 10146 2

no real roots

0106xx2

10436

4acb2 9146 2

one

096xx2

9436 4036

4

3636 0

State the value of the discriminant. Then determine the number of real roots of the equation.

9-A13 Skills Practice Wkb. Page 60 #8-24.