Radical Equations

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

An equation in which a variable occurs in the radicand 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 - 3 = 0 x ≥ 3 √ x - 3 - 3 √ x - 3 = 3 (√ x - 3 )2 = (3)2 √ 12 - 3 - 3 3 - 3 x - 3 = 9 x = 12 Therefore, the solution is x = 12.

010607b RADICALS: Solving Radicals 4

080602b RADICALS: Solving Radicals

≠ Solving Radical Equations 4 + √ 4 + x2 = x Check: x √ 4 + x2 = x - 4 Since the solution of is extraneous. Therefore, there are no real roots. x = ≠

080104b RADICALS: Solving Radicals 9

010305x RADICALS: Solving Radicals 11

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. 2x + 4 = x + 7 x = 3 Simplify. L.S. R.S. Verify your solution. Therefore, the solution is x = 3.

Squaring a Binomial (a + 2)2 = a2 + 4a + 4 ( 5 + √x - 2 )2 (a√x + b)2 Note that the middle term is twice the product of the two terms of the binomial. (a + 2)2 = a2 + 4a + 4 ( 5 + √x - 2 )2 The middle term will be twice the product of the two terms. A final concept that you should know: (a√x + b)2 = a2(x + b) = a2x + ab

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

x - 3 = 0 or x - 7 = 0 x = 3 or x = 7 Solving Radical Equations 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. L.S. R.S. L.S. R.S.

You must square the whole side NOT each term. One more to see another extraneous solution: a solution that you find algebraically but DOES NOT make a true statement when you substitute it back into the equation. The radical is already isolated 2 2 Square both sides You must square the whole side NOT each term. This must be FOILed You MUST check these answers Since you have a quadratic equation (has an x2 term) get everything on one side = 0 and see if you can factor this Doesn't work! Extraneous It checks!

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

Graph Graphing a Radical Function 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.

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 7x - 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 Solve x > -1 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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