How Do We Solve Radical Equations Do Now

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

Radical Equations 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. Check: L. S. R. S. Solve: √ x - 3 = 0 x≥ 3 0 √x-3 -3 √x-3 =3 (√ x - 3 )2 = (3)2 √ 12 - 3 3 -3 x-3=9 0 x = 12 Therefore, the solution is x = 12.

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

x = -1 is an extraneous solution.

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

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

Solving Radical Equations Solve 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. Use Foil. Simplify by dividing by a common factor of 2. Square both sides of the equation. Use Foil.

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

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

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

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

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

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

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

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

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