11 5 Solving Radical Equations Preview Warm Up

11 -5 Solving Radical Equations Preview Warm Up California Standards Lesson Presentation

11 -5 Solving Radical Equations Warm Up Solve each equation. 1. 3 x +5 = 17 4 2. 4 x + 1 = 2 x – 3 3. – 2 35 4. (x + 7)(x – 4) = 0 – 7, 4 5. x 2 – 11 x + 30 = 0 6, 5 6. x 2 = 2 x + 15 5, – 3

11 -5 Solving Radical Equations California Standards Extension of 2. 0 Students understand use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand use the rules of exponents.

11 -5 Solving Radical Equations Vocabulary radical equation

11 -5 Solving Radical Equations A radical equation is an equation that contains a variable within a radical. In this chapter, you will study radical equations that contain only square roots. Recall that you use inverse operations to solve equations. For nonnegative numbers, squaring and taking the square root are inverse operations. When an equation contains a variable within a square root, you can solve by squaring both sides of the equation.

11 -5 Solving Radical Equations =

11 -5 Solving Radical Equations Additional Example 1 A: Solving Simple Radical Equations Solve the equation. Check your answer. Square both sides. x = 25 Check 5 5 Substitute 25 for x in the original equation. 5 Simplify.

11 -5 Solving Radical Equations Additional Example 1 B: Solving Simple Radical Equations Solve the equation. Check your answer. Square both sides. 100 = 2 x 50 = x Divide both sides by 2. Check Substitute 50 for x in the original equation. 10 10 Simplify.

11 -5 Solving Radical Equations Check It Out! Example 1 a Solve the equation. Check your answer. Square both sides. Simplify. Check 6 6 Substitute 36 for x in the original equation. Simplify.

11 -5 Solving Radical Equations Check It Out! Example 1 b Solve the equation. Check your answer. Square both sides. 81 = 27 x 3=x Divide both sides by 27. Check Substitute 3 for x in the original equation. Simplify.

11 -5 Solving Radical Equations Check It Out! Example 1 c Solve the equation. Check your answer. Square both sides. 3 x = 1 Divide both sides by 3. Check Substitute for x in the original equation. Simplify.

11 -5 Solving Radical Equations Check It Out! Example 1 d Solve the equation. Check your answer. Square both sides. x = 12 Multiply both sides by 3.

11 -5 Solving Radical Equations Check It Out! Example 1 d continued Solve the equation. Check your answer. Check Substitute 12 for x. Simplify. 2 2

11 -5 Solving Radical Equations Some square-root equations do not have the square root isolated. To solve these equations, you may have to isolate the square root before squaring both sides. You can do this by using one or more inverse operations.

11 -5 Solving Radical Equations Additional Example 2 A: Solving Simple Radical Equations Solve the equation. Check your answer. Add 4 to both sides. Square both sides. x = 81 Check 9– 4 5 5 5

11 -5 Solving Radical Equations Additional Example 2 B: Solving Simple Radical Equations Solve the equation. Check your answer. Square both sides. x = 46 Subtract 3 from both sides. Check 7 7

11 -5 Solving Radical Equations Additional Example 2 C: Solving Simple Radical Equations Solve the equation. Check your answer. Subtract 6 from both sides. Square both sides. 5 x + 1 = 16 5 x = 15 x=3 Subtract 1 from both sides. Divide both sides by 5.

11 -5 Solving Radical Equations Additional Example 2 C Continued Solve the equation. Check your answer. Check 4+6 10 10 10

11 -5 Solving Radical Equations Check It Out! Example 2 a Solve the equation. Check your answer. Add 2 to both sides. Square both sides. x=9 Check 1 1

11 -5 Solving Radical Equations Check It Out! Example 2 b Solve the equation. Check your answer. Square both sides. x = 18 Subtract 7 from both sides. Check 5 5

11 -5 Solving Radical Equations Check It Out! Example 2 c Solve the equation. Check your answer. Add 1 to both sides. Square both sides. 3 x = 9 x=3 Subtract 7 from both sides. Divide both sides by 3.

11 -5 Solving Radical Equations Check It Out! Example 2 c Continued Solve the equation. Check your answer. Check 3 3

11 -5 Solving Radical Equations Additional Example 3 A: Solving Radical Equations by Multiplying or Dividing Solve the equation. Check your answer. Method 1 Divide both sides by 4. Square both sides. x = 64

11 -5 Solving Radical Equations Additional Example 3 A Continued Solve the equation. Check your answer. Method 2 Square both sides. x = 64 Divide both sides by 16.

11 -5 Solving Radical Equations Additional Example 3 A Continued Solve the equation. Check your answer. Check Substitute 64 for x in the original equation. 32 32 Simplify.

11 -5 Solving Radical Equations Additional Example 3 B: Solving Radical Equations by Multiplying or Dividing Solve the equation. Check your answer. Method 1 Multiply both sides by 2. Square both sides. 144 = x

11 -5 Solving Radical Equations Additional Example 3 B Continued Solve the equation. Check your answer. Method 2 Square both sides. Multiply both sides by 4. 144 = x

11 -5 Solving Radical Equations Additional Example 3 B Continued Solve the equation. Check your answer. Check Substitute 144 for x in the original equation. Simplify. 6 6

11 -5 Solving Radical Equations Check It Out! Example 3 a Solve the equation. Check your answer. Method 1 Divide both sides by 2. Square both sides.

11 -5 Solving Radical Equations Check It Out! Example 3 a Continued Solve the equation. Check your answer. Method 2 Square both sides. x = 121 Divide both sides by 4.

11 -5 Solving Radical Equations Check It Out! Example 3 a Continued Solve the equation. Check your answer. Check Substitute 121 for x in the original equation. Simplify.

11 -5 Solving Radical Equations Check It Out! Example 3 b Solve the equation. Check your answer. Method 1 Multiply both sides by 4. Square both sides. 64 = x

11 -5 Solving Radical Equations Check It Out! Example 3 b Continued Solve the equation. Check your answer. Method 2 Square both sides. Multiply both sides by 16.

11 -5 Solving Radical Equations Check It Out! Example 3 b Continued Solve the equation. Check your answer. Check Substitute 64 for x in the original equation. Simplify.

11 -5 Solving Radical Equations Check It Out! Example 3 c Solve the equation. Check your answer. Method 1 Multiply both sides by 5. Square both sides. Divide both sides by 4. x = 100

11 -5 Solving Radical Equations Check It Out! Example 3 c Continued Solve the equation. Check your answer. Method 2 Square both sides. 4 x = 400 x = 100 Multiply both sides by 25. Divide both sides by 4.

11 -5 Solving Radical Equations Check It Out! Example 3 c Continued Solve the equation. Check your answer. Check 4 4 Substitute 100 for x in the original equation. 4 Simplify.

11 -5 Solving Radical Equations Additional Example 4 A: Solving Radical Equations with Square Roots on Both Sides Solve the equation. Check your answer. Square both sides. 2 x – 1 = x + 7 x=8 Check Add 1 to both sides and subtract x from both sides.

11 -5 Solving Radical Equations Additional Example 4 B: Solving Radical Equations with Square Roots on Both Sides Solve the equation. Check your answer. Add to both sides. Square both sides. 5 x – 4 = 6 5 x = 10 x=2 Add 4 to both sides. Divide both sides by 5.

11 -5 Solving Radical Equations Additional Example 4 B Continued Solve the equation. Check your answer. Check 0 0

11 -5 Solving Radical Equations Check It Out! Example 4 a Solve the equation. Check your answer. Square both sides. 2 x = 4 x=2 Subtract x from both sides and subtract 2 from both sides. Divide both sides by 2.

11 -5 Solving Radical Equations Check It Out! Example 4 a Continued Solve the equation. Check your answer. Check

11 -5 Solving Radical Equations Check It Out! Example 4 b Solve the equation. Check your answer. Add to both sides. Square both sides. 2 x – 5 = 6 2 x = 11 Add 5 to both sides. Divide both sides by 2.

11 -5 Solving Radical Equations Check It Out! Example 4 b Continued Solve the equation. Check your answer. Check 0 0

11 -5 Solving Radical Equations Squaring both sides of an equation may result in an extraneous solution. Suppose your original equation is x = 3. Square both sides. Now you have a new equation. Solve this new equation for x by taking the square root of both sides. x=3 x 2 = 9 x = 3 or x = – 3

11 -5 Solving Radical Equations Now there are two solutions of the new equation. One (x = 3) is the original equation. The other (x = – 3) is extraneous–it is not a solution of the original equation. Because of extraneous solutions, it is especially important to check your answers to radical equations.

11 -5 Solving Radical Equations Additional Example 5 A: Extraneous Solutions Solve Check your answer. Subtract 12 from each sides. Square both sides 6 x = 36 x=6 Divide both sides by 6.

11 -5 Solving Radical Equations Additional Example 5 A Continued Solve Check your answer. Check Substitute 6 for x in the equation. 18 6 does not check; Ø. 6

11 -5 Solving Radical Equations Additional Example 5 B: Extraneous Solutions Solve Check your answer. Square both sides x 2 = 2 x + 3 x 2 – 2 x – 3 = 0 (x – 3)(x + 1) = 0 x – 3 = 0 or x + 1 = 0 x = 3 or x = – 1 Write in standard form. Factor. Zero-Product Property Solve for x.

11 -5 Solving Radical Equations Additional Example 5 B Continued Solve Check your answer. Check Substitute – 1 for x in the equation. – 1 1 Substitute 3 for x in the equation. 3 3 – 1 does not check; it is extraneous. The only solution is 3.

11 -5 Solving Radical Equations Check It Out! Example 5 a Solve the equation. Check your answer. Subtract 11 from both sides. Square both sides. x=5 Simplify.

11 -5 Solving Radical Equations Check It Out! Example 5 a Continued Solve the equation. Check your answer. Check Substitute 5 for x in the equation. 16 6 The answer is extraneous.

11 -5 Solving Radical Equations Check It Out! Example 5 b Solve the equation. Check your answer. Square both sides x 2 = – 3 x – 2 x 2 + 3 x + 2 = 0 (x + 1)(x + 2) = 0 x + 1 = 0 or x + 2 = 0 x = – 1 or x = – 2 Write in standard form. Factor. Zero-Product Property Solve for x.

11 -5 Solving Radical Equations Check It Out! Example 5 b Continued Solve the equation. Check your answer. Check Substitute – 1 for x in the equation. Substitute – 2 for x in the equation. – 2 2 Both answers are extraneous.

11 -5 Solving Radical Equations Check It Out! Example 5 c Solve the equation. Check your answer. Square both sides. x 2 – 4 x + 4 = x Subtract x from both sides. x 2 – 5 x + 4 = 0 (x – 1)(x – 4) = 0 x – 1 = 0 or x – 4 = 0 x = 1 or x = 4 Factor. Zero-Product Property Solve for x.

11 -5 Solving Radical Equations Check It Out! Example 5 c Continued Solve the equation. Check your answer. Check 2 2 Substitute 1 for x in the equation. Substitute 4 for x in the equation. 1 does not check; it is extraneous. The only solution is 4.

11 -5 Solving Radical Equations Additional Example 6: Geometry Application A triangle has an area of 36 square feet, its base is 8 feet, and its height is feet. What is the value of x? What is the height of the triangle? 8 ft Use the formula for area of a triangle. Substitute 8 for b, 36 for A, and for h. Simplify. Divide both sides by 4.

11 -5 Solving Radical Equations Additional Example 6 Continued A triangle has an area of 36 square feet, its base is 8 feet, and its height is feet. What is the value of x? What is the height of the triangle? Square both sides. 81 = x – 1 82 = x 8 ft

11 -5 Solving Radical Equations Additional Example 6 Continued A triangle has an area of 36 square feet, its base is 8 feet, and its height is feet. What is the value of x? What is the height of the triangle? 8 ft Check Substitute 82 for x. 36 36 The value of x is 82. The height of the triangle is 9 feet.

11 -5 Solving Radical Equations Check It Out! Example 6 A rectangle has an area of 15 cm 2. Its width is 5 cm, and its length is ( ) cm. What is the value of x? What is the length of the rectangle? A = lw 5 Use the formula for area of a rectangle. Substitute 5 for w, 15 for A, and for l. Divide both sides by 5.

11 -5 Solving Radical Equations Check It Out! Example 6 Continued A rectangle has an area of 15 cm 2. Its width is 5 cm, and its length is ( ) cm. What is the value of x? What is the length of the rectangle? 5 Square both sides. 8=x The value of x is 8. The length of the rectangle is cm.

11 -5 Solving Radical Equations Check It Out! Example 6 Continued A rectangle has an area of 15 cm 2. Its width is 5 cm, and its length is ( ) cm. What is the value of x? What is the length of the rectangle? Check A = lw Substitute 8 for x. 15 15 5

11 -5 Solving Radical Equations Lesson Quiz Solve each equation. Check your answer. 1. 2. 36 ø 3. 5. 45 4. 4 6. 11 4 7. A triangle has an area of 48 square feet, its base is 6 feet, and its height is feet. What is the value of x? What is the height of the triangle? 253; 16 ft