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Big Ideas Math Algebra 2, 2014

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Big Ideas Math Algebra 2, 2014 View details

4. Solving Radical Equations and Inequalities

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Exercise 7 Page 264

Raise each side of the equation to a power equal to the index of the radical to eliminate the radical.

x=3

Practice makes perfect

Solving a radical equation usually involves three main steps.

Isolate the radical on one side of the equation.
Raise each side of the equation to a power equal to the index of the radical to eliminate the radical.
Solve the resulting equation. Remember to check your results!

Now we can analyze the given radical equation. sqrt(x+6)-2=sqrt(x-2)Notice that in this equation there is an isolated radical with index equal to 2 on the right-hand side. Then, we will raise each side of the equation to the power of 2.

sqrt(x+6)-2=sqrt(x-2)

LHS^2=RHS^2

(sqrt(x+6)-2)^2=(sqrt(x-2))^2

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Simplify

( sqrt(a) )^2 = a

(sqrt(x+6)-2)^2=x-2

ExpandNegPerfectSquare

(a-b)^2=a^2-2ab+b^2

(sqrt(x+6))^2-2(sqrt(x+6))(2)+2^2=x-2

( sqrt(a) )^2 = a

x+6-2(sqrt(x+6))(2)+2^2=x-2

Multiply

x+6-4sqrt(x+6)+2^2=x-2

Calculate power

x+6-4sqrt(x+6)+4=x-2

Add terms

x+10-4sqrt(x+6)=x-2

We obtained another radical equation. Let's now isolate the radical, sqrt(x+6), on one side of the equation.

x+10-4sqrt(x+6)=x-2

LHS-10=RHS-10

x-4sqrt(x+6)=x-12

LHS-x=RHS-x

- 4sqrt(x+6)=- 12

.LHS /(- 4).=.RHS /(- 4).

sqrt(x+6)=3

Now, just like before, we will raise each side of the equation to the power of 2.

sqrt(x+6)=3

LHS^2=RHS^2

x+6=9

LHS-6=RHS-6

x=3

Next, we will check the solution by substituting 3 for x into the original equation. If the substitution produces a true statement, we know that our answer is correct. If it does not, then it is an extraneous solution.

sqrt(x+6)-2=sqrt(x-2)

x= 3

sqrt(3+6)-2? =sqrt(3-2)

▼

Simplify

Add and subtract terms

sqrt(9)-2? =sqrt(1)

Calculate root

3-2? =1

Subtract terms

1=1 ✓

Because our substitution produced a true statement, we know that our answer, x=3, is correct.

Subchapter links

Communicate Your Answer p.261

Monitoring Progress p.262-265

Exercises p.266-268