Solve radical equations and inequalities.

Objective Solve radical equations and inequalities.

Vocabulary radical equation radical inequality

A radical equationcontains 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.

Remember! For a square root, the index of the radical is 2.

Example 1A: Solving Equations Containing One Radical Solve each equation.

Example 1B: Solving Equations Containing One Radical Solve each equation.

Check It Out! Example 1a Solve the equation.

Check It Out! Example 1b Solve the equation.

Check It Out! Example 1c Solve the equation.

Solve Example 2: Solving Equations Containing Two Radicals

Check It Out! Example 2a Solve each equation.

Check It Out! Example 2b Solve each equation.

Helpful Hint You can use the intersect feature on a graphing calculator to find the point where the two curves intersect. Raising each side of an equation to an even power may introduce extraneous solutions.

Let Y1 = and Y2 = 5 – x. Example 3: Solving Equations with Extraneous Solutions Solve . Method 1 Use a graphing calculator. The graphs intersect in only one point, so there is exactly one solution. The solution is x = – 1

Example 3 Continued Method 2 Use algebra to solve the equation. Step 1 Solve for x. Square both sides. Simplify. –3x + 33 = 25 – 10x + x2 0 = x2 – 7x – 8 Write in standard form. 0 = (x – 8)(x + 1) Factor. x – 8 = 0 or x + 1 = 0 Solve for x. x = 8 or x = –1

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.

Check It Out! Example 3a Solve each equation. Method 1 Use a graphing calculator.

Let Y1 = and Y2 = x +3. Check It Out! Example 3a Solve each equation. Method 1 Use a graphing calculator. The graphs intersect in only one point, so there is exactly one solution. The solution is x = 1.

Check It Out! Example 3a Continued Method 2 Use algebra to solve the equation.

Check It Out! Example 3a Continued Method 1 Use algebra to solve the equation. Step 2 Use substitution to check for extraneous solutions. x 2 –2 4 4  Because x = –5 is extraneous, the only solution is x = 1.

Check It Out! Example 3b Solve each equation.

Let Y1 = and Y2 = –x +4. Check It Out! Example 3b Solve each equation. Method 1 Use a graphing calculator. The graphs intersect in two points, so there are two solutions. The solutions are x = –4 and x = 3.

Check It Out! Example 3b Continued Method 2 Use algebra to solve the equation.

Check It Out! Example 3b Continued Method 1 Use algebra to solve the equation. Step 2 Use substitution to check for extraneous solutions.  

Remember! To find a power, multiply the exponents.

1 3 Example 4A: Solving Equations with Rational Exponents Solve each equation. (5x + 7) = 3

2x = (4x + 8) 1 2 Example 4B: Solving Equations with Rational Exponents

1 1 1 1 1 1 2 2 2 2 2 2 2x = (4x + 8) 2x = (4x + 8) 2(–1) (4(–1) + 8) 4 4 –2 2 x  –2 4 4 16 2(2) (4(2) + 8) Example 4B Continued Step 2 Use substitution to check for extraneous solutions. The only solution is x = 2.

1 3 Check It Out! Example 4a Solve each equation. (x + 5) = 3

1 2 Check It Out! Example 4b (2x + 15) = x

1 1 1 1 1 1 2 2 2 2 2 2 (2x + 15) = x (2x + 15) = x (2(5) + 15) 5 (2(–3) + 15) –3 3 –3 x 5 5  (10 + 15) 5 (–6 + 15) –3 Check It Out! Example 4b Continued Step 2 Use substitution to check for extraneous solutions. The only solution is x = 5.

1 2 Check It Out! Example 4c 3(x + 6) = 9

Remember! A radical expression with an even index and a negative radicand has no real roots. A radical inequality is an inequality that contains a variable within a radical. You can solve radical inequalities by graphing or using algebra.

Solve . Example 5: Solving Radical Inequalities

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. Example 5: Solving Radical Inequalities Method 1 Use a graph and a table. The solution is 3 ≤ x ≤ 21.

Example 5 Continued Method 2 Use algebra to solve the inequality.

Solve . Check It Out! Example 5a

Solve . 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. Check It Out! Example 5a Method 1 Use a graph and a table. The solution is 3 ≤ x ≤ 12.

Check It Out! Example 5a Continued Method 2 Use algebra to solve the inequality.

Check It Out! Example 5b Solve .

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. Check It Out! Example 5b Solve . Method 1 Use a graph and a table. The solution is x ≥ –1.

Check It Out! Example 5b Continued Method 2 Use algebra to solve the inequality.

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.

Lesson Quiz: Part I Solve each equation or inequality. 1. 2. 3. 4. 5.

Solve radical equations and inequalities.