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

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

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11-5 Solving Radical Equations Extension of 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. California Standards

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11-5 Solving Radical Equations radical equation Vocabulary

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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.

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11-5 Solving Radical Equations =

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

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

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

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

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

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

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

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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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

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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.

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

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

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

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

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

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

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

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

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

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

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11-5 Solving Radical Equations Additional Example 6: Geometry Application 8 ft Use the formula for area of a triangle. Substitute 8 for b, 36 for A, and for h. Divide both sides by 4. 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? Simplify.

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11-5 Solving Radical Equations Additional Example 6 Continued 8 ft 82 = x Square both sides. 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? 81 = x – 1

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11-5 Solving Radical Equations 8 ft Check 36 The value of x is 82. The height of the triangle is 9 feet. Substitute 82 for x. 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?

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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? 5 A = lw Use the formula for area of a rectangle. Divide both sides by 5. Substitute 5 for w, 15 for A, and for l.

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11-5 Solving Radical Equations Check It Out! Example 6 Continued 8 = x Square both sides. 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 The value of x is 8. The length of the rectangle is cm.

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11-5 Solving Radical Equations CheckA = lw 15 Substitute 8 for x. 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

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11-5 Solving Radical Equations Lesson Quiz Solve each equation. Check your answer ø 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

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