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How Do We Solve Radical Equations? Do Now: Simplify the given expression. Do Now: Simplify the given expression. 1. 2. 1. 2.

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An equation in which a variable occurs in the radicand is called a radical equation. It should be noted, that when solving a radical equation algebraically, extraneous roots may be introduced when both sides of an equation are squared. Therefore, you must check your solutions for a radical equation. Solve: √ x - 3 - 3 = 0 √ x - 3 = 3 (√ x - 3 ) 2 = (3) 2 x - 3 = 9 x = 12 Check: √ x - 3 - 3 √ 12 - 3 - 3 3 - 3 0 0 Therefore, the solution is x = 12. x ≥ 3 Radical Equations L.S. R.S.

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4 + √ 4 + x 2 = x √ 4 + x 2 = x - 4 4 + x 2 = x 2 - 8x + 16 8x = 12 x Sincethe solution of x = is extraneous. Therefore, there are no real roots. Check: ≠ (√ 4 + x 2 ) 2 = (x - 4) 2 Solving Radical Equations

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x = -1 is an extraneous solution.

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Set up the equation so that there will be one radical on each side of the equal sign. Square both sides. Simplify. 2x + 4 = x + 7 x = 3 Verify your solution. Therefore, the solution is x = 3. x ≥ -2Solve Solving Radical Equations L.S. R.S.

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Squaring a Binomial (a + 2) 2 = a 2 + 4a + 4 Note that the middle term is twice the product of the two terms of the binomial. (a√x + b) 2 ( 5 + √x - 2 ) 2 The middle term will be twice the product of the two terms. A final concept that you should know: = a 2 x + ab = a 2 (x + b)

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Set up the equation so that there will be only one radical on each side of the equal sign. Square both sides of the equation. Simplify. Simplify by dividing by a common factor of 2. Square both sides of the equation. Use Foil. Solve Solving Radical Equations

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Distribute the 4. Simplify. Factor the quadratic. Solve for x. x - 3 = 0 or x - 7 = 0 x = 3 or x = 7 Verify both solutions. Solving Radical Equations L.S. R.S.

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One more to see another extraneous solution: The radical is already isolated 2 2 You must square the whole side NOT each term. Square both sides Since you have a quadratic equation (has an x 2 term) get everything on one side = 0 and see if you can factor this You MUST check these answers This must be FOILed Doesn't work! Extraneous It checks! a solution that you find algebraically but DOES NOT make a true statement when you substitute it back into the equation.

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Let's try another one: First isolate the radical - 1 3 3 Now since it is a 1/3 power this means the same as a cube root so cube both sides Now solve for x - 1 Let's check this answer It checks! Remember that the 1/3 power means the same thing as a cube root.

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Graph The domain is x > -2. The range is y > 0. Graphing a Radical Function

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Solve The solution will be the intersection of the graph and the graph of y = 0. The solution is x = 12. Check: 0 Solving a Radical Equation Graphically L.S. R.S.

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The solution is x = 3 or x = 7. Solve Solving a Radical Equation Graphically

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Find the values for which the graph of is above the graph of y = 3. The graphs intersect at x = 4. Note the radical 7x - 3 is defined only when. Therefore, the solution is x > 4. x > 4 Solving Radical Inequalities Solve

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The solution is -1 < x and x < 8. The graphs intersect at the point where x = 8. x ≥ -1 and x < 8 x > -1 Solving Radical Inequalities Solve

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