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Aim: Radical Equations Course: Alg. 2 & Trigonometry Aim: How do we solve radical equations? Do Now: Describe the steps for solving: x 2 – 80 = 0 x 2 = 80 Describe the reverse process add 80 to both sides take square root of both sides simplify x 2 – 80 = 0 subtract 80 from both sides square both sides x 2 = 80 solve by first squaring both sides. How do we solve?

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Aim: Radical Equations Course: Alg. 2 & Trigonometry Perfect Squares 12 144 11 121 100 10 9 9 81 8 8 64 7 7 49 6 6 36 5 5 25 4 4 16 3 3 9 4 2 2 1 1 1

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Aim: Radical Equations Course: Alg. 2 & Trigonometry Simplifying Radicals KEY: Find 2 factors for the radicand - one of which is the largest perfect square possible Multiplying Radicals

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Aim: Radical Equations Course: Alg. 2 & Trigonometry Dividing Radicals If quotient is not a perfect square you must simplify the radicand.

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Aim: Radical Equations Course: Alg. 2 & Trigonometry Adding/Subtracting Radicals Must have same radicand and index Add or subtract coefficients and combine result with the common radical Combined Result Coefficient Common Radical Unlike radicals must first be simplified to obtain like radicals (same radicand-same index), if possible. ex.

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Aim: Radical Equations Course: Alg. 2 & Trigonometry Solve and check: Isolate the radical: (already done) Solving Radical Equations Square each side: Solve the derived equation: x 2 – 9x + 16 = 0 use quadratic formula:

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Aim: Radical Equations Course: Alg. 2 & Trigonometry Solve and check: Isolate the radical: (already done) Solving Radical Equations Square each side: Solve the derived equation: x – 2 = 25 x = 27 Check: 5 = 5 (x – 2) 1/2 = 5 [(x – 2) 1/2 ] 2 = 5 2 alternate:

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Aim: Radical Equations Course: Alg. 2 & Trigonometry Solve and check: Extraneous Roots Isolate the radical: Square each side: Solve the derived equation: 2y – 1 = 9 y = 5 Check: 2y = 10 3 + 7 = 4 ? y = 5 is an extraneous root; there is no solution!

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Aim: Radical Equations Course: Alg. 2 & Trigonometry Solve and check: Solving Radical Equations Isolate the radical: Square each side: Solve the derived equation: Check each root: x = -1 is an extraneous root x 2 – 2x + 1 = x + 5 x 2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 x = 4 x = -1 4 = 4 ?

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Aim: Radical Equations Course: Alg. 2 & Trigonometry ? Solving Radical Equations Solve and check: Square each side: Solve the derived equation: 3 2 (x – 2) = 2 2 (x + 8) 9(x – 2) = 4(x + 8) 9x – 18 = 4x + 32 x = 10 x = 10 checks out as the solution

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Aim: Radical Equations Course: Alg. 2 & Trigonometry Evaluate for 500: Evaluate for 545: Model Problem The radical function is an approximation of the height in meters of a female giraffe using her weight x in kilograms. Find the heights of female giraffes with weights of 500 kg. and 545 kg. 3.17 m. 3.27 m.

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Aim: Radical Equations Course: Alg. 2 & Trigonometry The equation gives the time T in seconds it takes a body with mass 0.5 kg to complete one orbit of radius r meters. The force F in newtons pulls the body toward the center of the orbit. a. It takes 2 s for an object to make one revolution with a force of 10 N (newtons). Find the radius of the orbit. b. Find the radius of the orbit if the force is 160 N and T = 2. Model Problem a.

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