§ 7.6 Radical Equations

Blitzer, Intermediate Algebra, 4e – Slide #79 Solving Radical Equations Solving Radical Equations Containing nth Roots 1) If necessary, arrange terms so that one radical is isolated on one side of the equation. 2) Raise both sides of the equation to the nth power to eliminate the nth root. 3) Solve the resulting equation. If this equation still contains radicals, repeat steps 1 and 2. 4) Check all proposed solutions in the original equation.

Blitzer, Intermediate Algebra, 4e – Slide #80 Solving Radical EquationsEXAMPLE Solve: SOLUTION 1) Isolate a radical on one side. The radical,, can be isolated by subtracting 11 from both sides. We obtain Simplify. 2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides.

Blitzer, Intermediate Algebra, 4e – Slide #81 Solving Radical Equations This is the equation from step 2. 3) Solve the resulting equation. CONTINUED Subtract 5 from both sides. Divide both sides by 2. 4) Check the proposed solution in the original equation. Check 10: false Therefore there is no solution to the equation. ? ? ?

Blitzer, Intermediate Algebra, 4e – Slide #82 Solving Radical EquationsEXAMPLE Solve: SOLUTION 1) Isolate a radical on one side. The radical,, can be isolated by subtracting 3x from both sides. We obtain Simplify. Use the special formula 2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides.

Blitzer, Intermediate Algebra, 4e – Slide #83 Solving Radical Equations Equation from step 2. 3) Solve the resulting equation. Because of the -term, the resulting equation is a quadratic equation. We need to write this quadratic equation in standard form. We can obtain zero on the left side by subtracting 3x and 7 from both sides. CONTINUED Subtract 3x and 7 from both sides. Factor out the GCF, 9. Divide both sides by 9. Factor the right side.

Blitzer, Intermediate Algebra, 4e – Slide #84 Solving Radical EquationsCONTINUED Set each factor equal to 0. Solve for x. 4) Check the proposed solutions in the original equation. Check -2:Check -1: The solution is -1. The solution set is {-1}. truefalse ? ? ?? ? ?

Blitzer, Intermediate Algebra, 4e – Slide #85 Solving Radical EquationsEXAMPLE Solve: SOLUTION 1) Isolate a radical on one side. The radical,, can be isolated by subtracting from both sides. We obtain Simplify. Use the special formula 2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides.

Blitzer, Intermediate Algebra, 4e – Slide #86 Solving Radical Equations Combine like terms. CONTINUED 1) Isolate a radical on one side. The radical, can be isolated by subtracting 20 + x from both sides and then dividing both sides by -8. We obtain 2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides. Square the 3 and the

Blitzer, Intermediate Algebra, 4e – Slide #87 Solving Radical EquationsCONTINUED 3) Solve the resulting equation. This is the equation from the last step. Subtract 4 from both sides. 3) Check the proposed solution in the original equation. Check 5: The solution is 5. The solution set is {5}. true ? ? ?

Blitzer, Intermediate Algebra, 4e – Slide #88 Solving Radical EquationsEXAMPLE Solve: SOLUTION Although we can rewrite the equation in radical form This is the given equation. it is not necessary to do so. Because, the equation involves a fourth root, we isolate the radical term – that is, the term with the rational exponent – and raise both sides to the 4 th power. Subtract 7 from both sides.

Blitzer, Intermediate Algebra, 4e – Slide #89 Solving Radical Equations Raise both sides to the 4 th power. CONTINUED Multiply exponents on the left sides and then simplify. Subtract 3 from both sides. Divide both sides by 2. Upon checking the proposed solution, 39, in the original equation, we find that it checks out and is a solution. Therefore the solution set is {39}.

Blitzer, Intermediate Algebra, 4e – Slide #90 Solving Radical EquationsEXAMPLE For each planet in our solar system, its year is the time it takes the planet to revolve once around the sun. The function models the number of Earth days in a planet’s year, f (x), where x is the average distance of the planet from the sun, in millions of kilometers. Use the function to solve the following problem. There are approximately 88 Earth days in the year of the planet Mercury. What is the average distance of Mercury from the sun? Use a calculator and round to the nearest million kilometers.

Blitzer, Intermediate Algebra, 4e – Slide #91 Solving Radical EquationsSOLUTION CONTINUED To find the average distance of Mercury from the sun, replace f (x) in the function with 88. This is the given equation. Replace f (x) with 88. Divide both sides by 0.2. Square both sides. Simplify.

Blitzer, Intermediate Algebra, 4e – Slide #92 Solving Radical EquationsCONTINUED Take the cube root of both sides. Simplify. The model indicates that the average distance, to the nearest million kilometers, that Mercury is from the sun is 58 million kilometers.