Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities 8-8
Solving Radical Equations
and Inequalities
Holt Algebra 2
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Warm Up Simplify each expression. Assume all variables are positive. Write each expression in radical form.
1. 2.
3. 4.
5.
6.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Solve radical equations and inequalities.
Objective
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
radical equation
radical inequality
Vocabulary
Holt Algebra 2
8-8 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 Algebra 2
8-8 Solving Radical Equations
and Inequalities
For a square root, the index of the radical is 2.
Remember!
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Solve each equation.
Example 1A: Solving Equations Containing One
Radical
Subtract 5.
Simplify.
Square both sides.
Solve for x.
Simplify.
Check
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
7
Example 1B: Solving Equations Containing One
Radical
Divide by 7.
Simplify.
Cube both sides.
Solve for x.
Simplify.
Check
Solve each equation.
3
7
7 5x - 7 84
7 =
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Solve the equation.
Subtract 4.
Simplify.
Square both sides.
Solve for x.
Simplify.
Check
Check It Out! Example 1a
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Cube both sides.
Solve for x.
Simplify.
Check
Check It Out! Example 1b
Solve the equation.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Divide by 6.
Square both sides.
Solve for x.
Simplify.
Check
Check It Out! Example 1c
42 42Solve the equation.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Example 2: Solving Equations Containing Two
Radicals
Square both sides.
Solve
Simplify.
Distribute.
Solve for x.
7x + 2 = 9(3x – 2)
7x + 2 = 27x – 18
20 = 20x
1 = x
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Example 2 Continued
Check
3 3
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Square both sides.
Solve each equation.
Simplify.
Solve for x.
8x + 6 = 9x
6 = x
Check It Out! Example 2a
Check
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Cube both sides.
Solve each equation.
Simplify.
Distribute.
x + 6 = 8(x – 1)
Check It Out! Example 2b
x + 6 = 8x – 8
14 = 7x
2 = x Check
2 2
Solve for x.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Raising each side of an equation to an even power may introduce extraneous solutions.
You can use the intersect feature on a graphing calculator to find the point where the two curves intersect.
Helpful Hint
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Solve .
Example 3: Solving Equations with Extraneous
Solutions
Method 1 Use a graphing calculator.
The solution is x = – 1
The graphs intersect in only one point, so there is exactly one solution.
Let Y1 = and Y2 = 5 – x.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Example 3 Continued
Method 2 Use algebra to solve the equation.
Step 1 Solve for x.
Square both sides.
Solve for x.
Factor.
Write in standard form.
Simplify. –3x + 33 = 25 – 10x + x2
0 = x2 – 7x – 8
0 = (x – 8)(x + 1)
x – 8 = 0 or x + 1 = 0
x = 8 or x = –1
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Example 3 Continued
Method 2 Use algebra to solve the equation.
Step 2 Use substitution to check for extraneous solutions.
3 –3 6 6 x
Because x = 8 is extraneous, the only solution is x = –1.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Method 1 Use a graphing calculator.
The solution is x = 1.
The graphs intersect in only one point, so there is exactly one solution.
Check It Out! Example 3a
Solve each equation.
Let Y1 = and Y2 = x +3.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Method 2 Use algebra to solve the equation.
Step 1 Solve for x.
Square both sides.
Solve for x.
Factor.
Write in standard form.
Simplify. 2x + 14 = x2 + 6x + 9
0 = x2 + 4x – 5
0 = (x + 5)(x – 1)
x + 5 = 0 or x – 1 = 0
x = –5 or x = 1
Check It Out! Example 3a Continued
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Method 1 Use algebra to solve the equation.
Step 2 Use substitution to check for extraneous solutions.
4 4 x
Because x = –5 is extraneous, the only solution is x = 1.
Check It Out! Example 3a Continued
2 14 35 5- - 2 –2
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Method 1 Use a graphing calculator.
Check It Out! Example 3b
Let Y1 = and Y2 = –x +4.
The solutions are x = –4 and x = 3.
The graphs intersect in two points, so there are two solutions.
Solve each equation.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Method 2 Use algebra to solve the equation.
Step 1 Solve for x.
Square both sides.
Solve for x.
Factor.
Write in standard form.
Simplify. –9x + 28 = x2 – 8x + 16
0 = x2 + x – 12
0 = (x + 4)(x – 3)
x + 4 = 0 or x – 3 = 0
x = –4 or x = 3
Check It Out! Example 3b Continued
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Method 1 Use algebra to solve the equation.
Step 2 Use substitution to check for extraneous solutions.
Check It Out! Example 3b Continued
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
To find a power, multiply the exponents.
Remember!
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Example 4A: Solving Equations with Rational
Exponents
Solve each equation.
(5x + 7) = 3
Cube both sides.
Solve for x.
Factor.
Write in radical form.
Simplify. 5x + 7 = 27
5x = 20
x = 4
1
3
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Example 4B: Solving Equations with Rational
Exponents
Factor.
Solve for x.
Factor out the GCF, 4.
Raise both sides to the
reciprocal power.
Simplify.
2x = (4x + 8) 1
2
(2x)2 = [(4x + 8) ]
2
1
2
4x2 = 4x + 8
Write in standard form. 4x2 – 4x – 8 = 0
4(x2 – x – 2) = 0
4(x – 2)(x + 1) = 0
4 ≠ 0, x – 2 = 0 or x + 1 = 0
x = 2 or x = –1
Step 1 Solve for x.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Example 4B Continued
Step 2 Use substitution to check for extraneous
solutions.
2x = (4x + 8) 1
2
2(2) (4(2) + 8) 1
2
4 16 1
2
4 4
2x = (4x + 8) 1
2
2(–1) (4(–1) + 8) 1
2
–2 4 1
2
–2 2 x
The only solution is x = 2.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Solve each equation.
(x + 5) = 3
Cube both sides.
Solve for x.
Write in radical form.
Simplify. x + 5 = 27
x = 22
1
3
Check It Out! Example 4a
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Factor.
Solve for x.
Raise both sides to the
reciprocal power.
Simplify.
(2x + 15) = x 1
2
[(2x + 15) ]2 =
(x)
2
1
2
2x + 15 = x2
Write in standard form. x2 – 2x – 15 = 0
(x – 5)(x + 3) = 0
x – 5 = 0 or x + 3 = 0
x = 5 or x = –3
Step 1 Solve for x.
Check It Out! Example 4b
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
The only solution is x = 5.
Check It Out! Example 4b Continued
(2(–3) + 15) –3 1
2
(–6 + 15) –3 1
2
x
(2x + 15) = x 1
2
3 –3
Step 2 Use substitution to check for extraneous
solutions.
(2(5) + 15) 5 1
2
(2x + 15) = x 1
2
(10 + 15) 5 1
2
5 5
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Solve for x.
Raise both sides to the
reciprocal power.
Simplify.
9(x + 6) = 81
[3(x + 6) ]2 = (9)
2
1
2
Distribute 9. 9x + 54 = 81
Check It Out! Example 4c
9x = 27
x = 3
3(x + 6) = 9 1
2
Simplify.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
A radical inequality is an inequality that contains a variable within a radical. You can solve radical inequalities by graphing or using algebra.
A radical expression with an even index and a negative radicand has no real roots.
Remember!
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Example 5: Solving Radical Inequalities
Method 1 Use a graph and a table.
Solve .
On a graphing calculator, let Y1 = and Y2 = 9. The graph of Y1 is at or below the graph of Y2 for values of x between 3 and 21. Notice that Y1 is undefined when < 3.
The solution is 3 ≤ x ≤ 21.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Example 5 Continued
Method 2 Use algebra to solve the inequality.
Step 1 Solve for x.
Subtract 3.
Solve for x.
Simplify.
Square both sides.
2x ≤ 42
x ≤ 21
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Example 5 Continued
Method 2 Use algebra to solve the inequality.
Step 2 Consider the radicand.
The radicand cannot be negative.
Solve for x.
2x – 6 ≥ 0
2x ≥ 6
x ≥ 3
The solution of is x ≥ 3 and x ≤ 21, or 3 ≤ x ≤ 21.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Method 1 Use a graph and a table.
Check It Out! Example 5a
On a graphing calculator, let Y1 = and Y2 = 5. The graph of Y1 is at or below the graph of Y2 for values of x between 3 and 12. Notice that Y1 is undefined when < 3.
The solution is 3 ≤ x ≤ 12.
Solve .
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Method 2 Use algebra to solve the inequality.
Step 1 Solve for x.
Subtract 2.
Solve for x.
Simplify.
Square both sides.
x – 3 ≤ 9
x ≤ 12
Check It Out! Example 5a Continued
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Method 2 Use algebra to solve the inequality.
Step 2 Consider the radicand.
The radicand cannot be negative.
Solve for x.
x – 3 ≥ 0
x ≥ 3
Check It Out! Example 5a Continued
The solution of is x ≥ 3 and x ≤ 12, or 3 ≤ x ≤ 12.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Method 1 Use a graph and a table.
Check It Out! Example 5b
Solve .
The solution is x ≥ –1.
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.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Method 2 Use algebra to solve the inequality.
Step 1 Solve for x.
Solve for x.
Cube both sides.
x + 2 ≥ 1
x ≥ –1
Check It Out! Example 5b Continued
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Method 2 Use algebra to solve the inequality.
Step 2 Consider the radicand.
The radicand cannot be negative.
Solve for x.
x + 2 ≥ 1
x ≥ –1
Check It Out! Example 5b Continued
The solution of is x ≥ –1.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Example 6: Automobile Application
The time t in seconds that it takes a car to
travel a quarter mile when starting from a
full stop can be estimated by using the
formula , where w is the weight
of the car in pounds and P is the power
delivered by the engine in horsepower. If
the quarter-mile time from a 3590 lb car is
13.4 s, how much power does its engine
deliver? Round to the nearest horsepower.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Example 6 Continued Use the formula to determine the amount of horsepower the 3590 lb car has if it finishes the quarter-mile in 13.4s.
Substitute 13.4 for t
and 3590 for w.
Cube both sides.
Solve for P.
Simplify.
2406.104P ≈ 709,548.747
P ≈ 295 The engine delivers a power of about 295 hp.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
A car skids to a stop on a street with a speed limit of 30 mi/h. The skid marks measure 35 ft, and the coefficient of friction was 0.7. Was the car speeding? Explain.
Check It Out! Example 6
Use the formula to determine the greatest possible length of the driver’s skid marks if he was traveling 30 mi/h.
The speed s in miles per hour that a car is traveling when it goes into a skid can be estimated by using the formula s = , where f is the coefficient of friction and d is the length of the skid marks in feet.
30 fd
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Check It Out! Example 6 Continued
900 = 21d
43 ≈ d
If the car were traveling 30 mi/h, its skid marks would have measured about 43 ft. Because the actual skid marks measure less than 43 ft, the car was not speeding.
Substitute 30 for s and 0.7 for f.
Simplify.
Square both sides.
Simplify.
Solve for d.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Lesson Quiz: Part I
Solve each equation or inequality.
x = 16 1.
2.
3.
4.
x = –5
x = 14
x = 2
x ≥ 59 5.
Holt Algebra 2
8-8 Solving Radical Equations
and Inequalities
Lesson Quiz: Part II
137,258 ft3
6. The radius r in feet of a spherical water tank
can be determined by using the formula ,
where V is the volume of the tank in cubic feet. To the nearest cubic foot, what is the volume of a spherical tank with a radius of 32 ft?