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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 26 Page 266

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

x=1/4

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

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

LHS^2=RHS^2

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

▼

Simplify

( sqrt(a) )^2 = a

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

ExpandNegPerfectSquare

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

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

Calculate power

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

Multiply

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

( sqrt(a) )^2 = a

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

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

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

LHS-2=RHS-2

x=2-4sqrt(x)+x

LHS-x=RHS-x

0=2-4sqrt(x)

LHS+4sqrt(x)=RHS+4sqrt(x)

4sqrt(x)=2

.LHS /4.=.RHS /4.

sqrt(x)=1/2

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

sqrt(x)=1/2

LHS^2=RHS^2

x=1/4

Next, we will check the solution by substituting 14 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+2)=2-sqrt(x)

x= 1/4

sqrt(1/4+2)? =2-sqrt(1/4)

▼

Simplify

a = 4* a/4

sqrt(1/4+8/4)? =2-sqrt(1/4)

Add fractions

sqrt(9/4)? =2-sqrt(1/4)

sqrt(a/b)=sqrt(a)/sqrt(b)

sqrt(9)/sqrt(4)? =2-sqrt(1)/sqrt(4)

Calculate root

3/2? =2-1/2

a = 2* a/2

3/2? =4/2-1/2

Subtract fractions

3/2=3/2 ✓

Because our substitution produced a true statement, we know that our answer, x= 14, is correct.

Subchapter links

Communicate Your Answer p.261

Monitoring Progress p.262-265

Exercises p.266-268