Solving
Radical
Equations
Solving
Radical
Equations
5-8
5-8 and
andInequalities
Inequalities
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
5-8
Solving Radical Equations
and Inequalities
Warm Up
Simplify each expression. Assume all
variables are positive.
1.
2.
3.
4.
Write each expression in radical form.
5.
6.
Holt McDougal Algebra 2
5-8
Solving Radical Equations
and Inequalities
Objective
Solve radical equations and inequalities.
Holt McDougal Algebra 2
5-8
Solving Radical Equations
and Inequalities
Vocabulary
radical equation
radical inequality
Holt McDougal Algebra 2
5-8
Solving Radical Equations
and Inequalities
Essential Question
• How do you solve a radical equation?
Holt McDougal Algebra 2
5-8
Solving Radical Equations
and Inequalities
A radical equation contains a variable within a
radical. Recall that you can solve quadratic
equations by taking the square root of both
sides. Similarly, radical equations can be solved
by raising both sides to a power.
Holt McDougal Algebra 2
5-8
Solving Radical Equations
and Inequalities
Remember!
For a square root, the index of the radical is 2.
Holt McDougal Algebra 2
5-8
Solving Radical Equations
and Inequalities
Example 1A: Solving Equations Containing One
Radical
Solve each equation.
Check
Isolate the radical.
Simplify.
Square both sides.
Simplify.
Solve for x.
Holt McDougal Algebra 2

5-8
Solving Radical Equations
and Inequalities
Example 1B: Solving Equations Containing One
Radical
Solve each equation.
Check
7 3 5x - 7 84
=
7
7
Isolate the radical.
7
Simplify.
Cube both sides.
Simplify.
Solve for x.
Holt McDougal Algebra 2

5-8
Solving Radical Equations
and Inequalities
Example 2: Solving Equations Containing Two
Radicals
Solve
Square both sides.
7x + 2 = 9(3x – 2)
Simplify.
7x + 2 = 27x – 18
Distribute.
20 = 20x
1=x
Holt McDougal Algebra 2
Solve for x.
5-8
Solving Radical Equations
and Inequalities
Example 2 Continued
Check
3
Holt McDougal Algebra 2
3 
5-8
Solving Radical Equations
and Inequalities
Check It Out! Example 2b
Solve each equation.
Cube both sides.
x + 6 = 8(x – 1)
Simplify.
x + 6 = 8x – 8
14 = 7x
2 =x
Distribute.
Solve for x.
Check
2
Holt McDougal Algebra 2
2 
5-8
Solving Radical Equations
and Inequalities
Raising each side of an equation to an even
power may introduce extraneous solutions.
Helpful Hint
You can use the intersect feature on a graphing
calculator to find the point where the two curves
intersect.
Holt McDougal Algebra 2
5-8
Solving Radical Equations
and Inequalities
Example 3: Solving Equations with Extraneous
Solutions
Solve
.
Method 2 Use a graphing
calculator.
Let Y1 =
and Y2 = 5 – x.
The graphs intersect in only one point,
so there is exactly one solution.
The solution is x = – 1
Holt McDougal Algebra 2
5-8
Solving Radical Equations
and Inequalities
Check It Out! Example 3a
Solve each equation.
Method 2 Use a graphing
calculator.
Let Y1 =
and Y2 = x +3.
The graphs intersect in only one point,
so there is exactly one solution.
The solution is x = 1.
Holt McDougal Algebra 2
5-8
Solving Radical Equations
and Inequalities
Check It Out! Example 3b
Solve each equation.
Method 2 Use a graphing
calculator.
Let Y1 =
and Y2 = –x +4.
The graphs intersect in two
points, so there are two
solutions.
The solutions are x = –4
and x = 3.
Holt McDougal Algebra 2
5-8
Solving Radical Equations
and Inequalities
Remember!
To find a power, multiply the exponents.
Holt McDougal Algebra 2
5-8
Solving Radical Equations
and Inequalities
Example 4A: Solving Equations with Rational
Exponents
Solve each equation.
(5x + 7)
1
3
=3
3


3
5
x

7
=
3



Cube both sides.
5x + 7 = 27
Simplify.
1
3
5x = 20
x= 4
Holt McDougal Algebra 2
Add -7 to both sides.
Solve for x.
5-8
Solving Radical Equations
and Inequalities
Lesson Quiz: Part I
Solve each equation.
1.
x = 16
2.
x = –5
3.
x = 14
Holt McDougal Algebra 2
5-8
Solving Radical Equations
and Inequalities
Essential Question
• How do you solve a radical equation?
Holt McDougal Algebra 2
5-8
Solving Radical Equations
and Inequalities
• Teacher: Your behavior reminds me of
2
Student: Why?
Teacher: Because it’s completely irrational.
Holt McDougal Algebra 2
5-8
Solving Radical Equations
and Inequalities
Check It Out! Example 5b
Solve
.
Method 1 Use a graph and a table.
On a graphing calculator, let Y1 =
and Y2 = 1.
The graph of Y1 is at or above the graph of Y2 for
values of x greater than –1. Notice that Y1 is
undefined when < –2.
The solution is x ≥ –1.
Holt McDougal Algebra 2