Aim: How do we solve radical equations?

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Presentation transcript:

Aim: How do we solve radical equations? Do Now: Describe the steps for solving: x2 – 80 = 0 x2 = 80 add 80 to both sides take square root of both sides simplify Describe the reverse process square both sides x2 = 80 x2 – 80 = 0 subtract 80 from both sides How do we solve? solve by first squaring both sides.

Perfect Squares 12 144 11 121 100 10 9 81 8 64 7 49 6 36 5 25 4 16 3 9 4 2 1

Simplifying Radicals KEY: Find 2 factors for the radicand - one of which is the largest perfect square possible Multiplying Radicals

Dividing Radicals If quotient is not a perfect square you must simplify the radicand.

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

Solving Radical Equations Solve and check: Isolate the radical: (already done) Square each side: Solve the derived equation: x2 – 9x + 16 = 0 use quadratic formula:

Solving Radical Equations Solve and check: Isolate the radical: (already done) Square each side: Solve the derived equation: x – 2 = 25 x = 27 Check: (x – 2)1/2 = 5 [(x – 2)1/2]2 = 52 alternate: 5 = 5

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

Solving Radical Equations Solve and check: Isolate the radical: Square each side: Solve the derived equation: x2 – 2x + 1 = x + 5 x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 x = 4 x = -1 Check each root: ? 4 = 4 x = -1 is an extraneous root

Solving Radical Equations Solve and check: ? Square each side: Solve the derived equation: 32(x – 2) = 22(x + 8) 9(x – 2) = 4(x + 8) 9x – 18 = 4x + 32 x = 10 x = 10 checks out as the solution

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. Evaluate for 500: Evaluate for 545:  3.17 m.  3.27m.

Model Problem 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. a.