### Equations with Radicals on Both Sides

Often equations have more than one radical expression. The strategy in this case is to start by isolating the most complicated radical expression and raise the equation to the appropriate power. We then repeat the process until all radical signs are eliminated.

#### Finding the Real Roots of an Equation

Find the real roots of the equation \begin{align*}\sqrt{2x+1}-\sqrt{x-3}=2\end{align*}.

\begin{align*}\text{Isolate one of the radical expressions:} && \sqrt{2x+1}& =2+\sqrt{x-3}\\ \text{Square both sides:} && \left(\sqrt{2x+1}\right)^2& =\left(2+\sqrt{x-3}\right)^2\\ \text{Eliminate parentheses:} && 2x+1& =4+4\sqrt{x-3}+x-3\\ \text{Simplify:} && x& =4 \sqrt{x-3}\\ \text{Square both sides of the equation:} && x^2& =\left(4 \sqrt{x-3} \right)^2\\ \text{Eliminate parentheses:} && x^2& =16(x-3)\\ \text{Simplify:} && x^2& =16x-48\\ \text{Move all terms to one side of the equation:} && x^2-16x+48& =0\\ \text{Factor:} && (x-12)(x-4)& =0\\ \text{Solve:} && x& =12 \ \text{or} \ x=4\end{align*}

**Check:** \begin{align*}\sqrt{2(12)+1}-\sqrt{12-3}=\sqrt{25}-\sqrt{9}=5-3=2.\end{align*} The solution checks out.

\begin{align*}\sqrt{2(4)+1}-\sqrt{4-3}=\sqrt{9}-\sqrt{1}=3-1=2\end{align*} The solution checks out.

The equation has two solutions: \begin{align*}x=12\end{align*} and \begin{align*}x=4\end{align*}.

**Identify Extraneous Solutions to Radical Equations**

We saw in Example 3 that some of the solutions that we find by solving radical equations do not check out when we substitute (or “plug in”) those solutions back into the original radical equation. These are called **extraneous solutions.** It is very important to check the answers we obtain by plugging them back into the original equation, so we can tell which of them are real solutions.

#### Solving for Unknown Values

Find the real solutions of the equation \begin{align*}\sqrt{x-3}-\sqrt{x}=1\end{align*}.

\begin{align*}\text{Isolate one of the radical expressions:} && \sqrt{x-3}&=\sqrt{x}+1\\ \text{Square both sides:} && \left(\sqrt{x-3}\right)^2& =\left(\sqrt{x}+1\right)^2\\ \text{Remove parenthesis:} && x-3& =\left(\sqrt{x}\right)^2+2\sqrt{x}+1\\ \text{Simplify:} && x-3& =x+2\sqrt{x}+1\\ \text{Now isolate the remaining radical:} && -4& =2\sqrt{x}\\ \text{Divide all terms by 2:} && -2& =\sqrt{x}\\ \text{Square both sides:} && x& =4\end{align*}

**Check:** \begin{align*}\sqrt{4-3}-\sqrt{4}=\sqrt{1}-2=1-2=-1\end{align*} The solution does not check out.

** The equation has no real solutions.** \begin{align*}x=4\end{align*} is an extraneous solution.

**Applications using Special Case of Radical Equations**

Radical equations often appear in problems involving areas and volumes of objects.

#### Real-World Application: Garden Fencing

Anita’s square vegetable garden is 21 square feet larger than Fred’s square vegetable garden. Anita and Fred decide to pool their money together and buy the same kind of fencing for their gardens. If they need 84 feet of fencing, what is the size of each garden?

**Make a sketch:**

**Define variables:** Let Fred’s area be \begin{align*}x\end{align*}; then Anita’s area is \begin{align*}x+21\end{align*}.

**Find an equation:**

Side length of Fred’s garden is \begin{align*}\sqrt{x}\end{align*}

Side length of Anita’s garden is \begin{align*}\sqrt{x+21}\end{align*}

The amount of fencing is equal to the combined perimeters of the two squares:

\begin{align*}4\sqrt{x}+4\sqrt{x+21}=84\end{align*}

**Solve the equation:**

\begin{align*}\text{Divide all terms by 4:} && \sqrt{x}+\sqrt{x+21}& =21\\ \text{Isolate one of the radical expressions:} && \sqrt{x+21}& =21-\sqrt{x}\\ \text{Square both sides:} && \left(\sqrt{x+21}\right)^2& =\left(21-\sqrt{x}\right)^2\\ \text{Eliminate parentheses:} && x+21& =441-42\sqrt{x}+x\\ \text{Isolate the radical expression:} && 42\sqrt{x}& =420\\ \text{Divide both sides by 42:} && \sqrt{x}& =10\\ \text{Square both sides:} && x& =100 \ ft^2\end{align*}

**Check:** \begin{align*}4\sqrt{100}+4\sqrt{100+21}=40+44=84\end{align*}. **The solution checks out.**

Fred’s garden is \begin{align*}10 \ ft \times 10 \ ft = 100 \ ft^2\end{align*} and Anita’s garden is \begin{align*}11 \ ft \times 11 \ ft = 121 \ ft^2\end{align*}.

### Example

#### Example 1

Find the real solutions of the equation \begin{align*}\sqrt{9-x}=3+\sqrt{2x}\end{align*}.

\begin{align*}\text{Isolate one of the radical expressions:} && \sqrt{9-x}&=3+\sqrt{2x}\\ \text{Square both sides:} && (\sqrt{9-x})^2&=(3+\sqrt{2x})^2\\ \text{Remove parenthesis:} && 9-x&=9+6\sqrt{2x}+2x\\ \text{Simplify:} && x& =-2\sqrt{2x}\\ \text{Square both sides:} && x^2& =4(2x)\\ \text{Simplify:} && x^2& =8x\\ \text{Set one side equal to zero:} && x^2-8x& =0\\ \text{Factor:} && x(x-8)& =0\\ \text{Use the zero product principle:} && x& =0 \text{ or } x=8\end{align*}

**Check:** First check \begin{align*}x=0\end{align*}:

\begin{align*}\text{Start with the original equation:} && \sqrt{9-x}&=3+\sqrt{2x}\\ \text{Substitute in the x-value:} && \sqrt{9-8}&=3+\sqrt{2(8)}\\ \text{Simplify:} && \sqrt{1}&=3+\sqrt{16)}\\ \text{Take the square root:} && 1&=3+4=7\\ \end{align*}

The solution does not check out.

Then check \begin{align*}x=8\end{align*}:

\begin{align*}\text{Start with the original equation:} && \sqrt{9-x}&=3+\sqrt{2x}\\ \text{Substitute in the x-value:} && \sqrt{9-0}&=3+\sqrt{2(0)}\\ \text{Simplify:} && \sqrt{9}&=3+\sqrt{0)}\\ \text{Take the square root:} && 3&=3+0=3\\ \end{align*}

The solution checks out.

** The equation has one real solution.** \begin{align*}x=8\end{align*} is an extraneous solution.

### Review

Find the solution to each of the following radical equations. Identify extraneous solutions.

- \begin{align*}\sqrt{x}=\sqrt{x-9}+1\end{align*}
- \begin{align*}\sqrt{x}+2=\sqrt{3x-2}\end{align*}
- \begin{align*}5 \sqrt{x}=\sqrt{x+12}+6\end{align*}
- \begin{align*}\sqrt{10-5x}+\sqrt{1-x}=7\end{align*}
- \begin{align*}\sqrt{2x-2}-2\sqrt{x}+2=0\end{align*}
- \begin{align*}\sqrt{2x+5}-3\sqrt{2x-3}=\sqrt{2-x}\end{align*}
- \begin{align*}3\sqrt{x}-9=\sqrt{2x-14}\end{align*}
- \begin{align*}\sqrt{x+7}=\sqrt{x+4}+1\end{align*}
- \begin{align*}\sqrt{4x}=\sqrt{3-2x}-1\end{align*}
- \begin{align*}\sqrt{2x+5}+\sqrt{3-x}=10\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 11.7.