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Radical Equations

Find the roots of basic equations containing roots

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Solving Simple Radical Equations

The legs of a right triangle measure 3 and \begin{align*}2\sqrt {x}\end{align*}. The hypotenuse measures 5. What is the length of the leg with the unknown value?

Solving Radical Equations

Solving radical equations are very similar to solving other types of equations. The objective is to get \begin{align*}x\end{align*} by itself. However, now there are radicals within the equations. Recall that the opposite of the square root of something is to square it.

Let's determine if \begin{align*}x = 5\end{align*} is the solution to \begin{align*}\sqrt{2x+15}=8\end{align*}.

Plug in 5 for \begin{align*}x\end{align*} to see if the equation holds true. If it does, then 5 is the solution.

\begin{align*}\sqrt{2 \left(5\right)+15}&=8 \\ \sqrt{10+15}&=9 \\ \sqrt{25} &\neq 8\end{align*}

We know that \begin{align*}\sqrt{25}=5\end{align*}, so \begin{align*}x = 5\end{align*} is not the solution. 

Now, let's solve the following equations for x.

  1. \begin{align*}\sqrt{2x-5}+7=16\end{align*}

To solve for \begin{align*}x\end{align*}, we need to isolate the radical. Subtract 7 from both sides.

\begin{align*}\sqrt{2x-5}+7&=16 \\ \sqrt{2x-5}&=9\end{align*}

Now, we can square both sides to eliminate the radical. Only square both sides when the radical is alone on one side of the equals sign.

\begin{align*}\sqrt{2x-5}^2&=9^2 \\ 2x-5&=81 \\ 2x&=86 \\ x&=43\end{align*}

Check: \begin{align*}\sqrt{2 \left(43\right)-5}+7=\sqrt{86-5}+7=\sqrt{81}+7=9+7=16\end{align*}

ALWAYS check your answers when solving radical equations. Sometimes, you will solve an equation, get a solution, and then plug it back in and it will not work. These types of solutions are called extraneous solutions and are not actually considered solutions to the equation. 

  1. \begin{align*}3\sqrt[3]{x-8}-2=-14\end{align*}

Again, isolate the radical first. Add 2 to both sides and divide by 3.

\begin{align*}3\sqrt[3]{x-8}-2&=-14\\ 3\sqrt[3]{x-8}&=-12\\ \sqrt[3]{x-8}&=-4\end{align*}

Now, cube both sides to eliminate the radical.

\begin{align*}\sqrt[3]{x-8}^3&=(-4)^3\\ x-8&=-64\\ x&=-56\end{align*}

Check: \begin{align*}3\sqrt[3]{-56-8}-2=3 \sqrt[3]{-64}-2=3 \cdot -4-2=-12-2=-14\end{align*}

Examples

Example 1

Earlier, you were asked to find the length of the leg with the unknown value. 

Use the Pythagorean Theorem and solve for x then substitute that value in to solve for the leg with the unknown.

\begin{align*}3^2 +(2\sqrt {x})^2) = 5^2\\ 9 + 4x = 25\\ 4x = 16\\ x = 4\end{align*}

Now substitute this value into the leg with the unknown.

\begin{align*}2 \sqrt{4} = 2 \cdot 2 = 4\end{align*}

Therefore the leg with the unknown has a length of 4.

Example 2

Solve for x: \begin{align*}\sqrt{x+5}=6\end{align*}. Check your answer.

The radical is already isolated here. Square both sides and solve for \begin{align*}x\end{align*}.

\begin{align*}\sqrt{x+5}^2&=6^2 \\ x+5&=36 \\ x&=31\end{align*}

Check: \begin{align*}\sqrt{31+5}=\sqrt{36}=6 \end{align*}

Example 3

Solve for x: \begin{align*}5\sqrt{2x-1}+1=26\end{align*}. Check your answer.

Isolate the radical by subtracting 1 and then dividing by 5.

\begin{align*}5\sqrt{2x-1}+1&=26 \\ 5\sqrt{2x-1}&=25 \\ \sqrt{2x-1}&=5\end{align*}

Square both sides and continue to solve for \begin{align*}x\end{align*}.

\begin{align*}\sqrt{2x-1}^2&=5^2 \\ 2x-1&=25 \\ 2x&=26 \\ x&=13\end{align*}

Check: \begin{align*}5\sqrt{2 \left(13\right)-1}+1=5\sqrt{26-1}=5\sqrt{25}+1=5 \cdot 5+1=25+1=26\end{align*}

Example 4

Solve for x: \begin{align*}\sqrt[4]{3x+11}-2=3\end{align*}. Check your answer.

In this problem, we have a fourth root. That means, once we isolate the radical, we must raise both sides to the fourth power to eliminate it.

\begin{align*}\sqrt[4]{3x+11}-2&=3\\ \sqrt[4]{3x-11}^4&=5^4\\ 3x-11&=625\\ 3x&=636\\ x&=212\end{align*}

Check: \begin{align*}\sqrt[4]{3 \left(212\right)+11}-2=\sqrt[4]{636-11}-2=\sqrt[4]{625}-2=5-2=3\end{align*}

Review

Determine if the given values of x are solutions to the radical equations below.

  1. \begin{align*}\sqrt{x-3}=7; x = 32\end{align*}
  2. \begin{align*}\sqrt[3]{6+x}=3; x = 21\end{align*}
  3. \begin{align*}\sqrt[4]{2x+3}-11=-9; x = 6\end{align*}

Solve the equations and check your answers.

  1. \begin{align*}\sqrt{x+5}=6\end{align*}
  2. \begin{align*}2- \sqrt{x+1}=0\end{align*}
  3. \begin{align*}4 \sqrt{5-x}=12\end{align*}
  4. \begin{align*}\sqrt{x+9}+7=11\end{align*}
  5. \begin{align*}\frac{1}{2}\sqrt[3]{x-2}=1\end{align*}
  6. \begin{align*}\sqrt[3]{x+3}+5=9\end{align*}
  7. \begin{align*}5\sqrt{15-x}+2=17\end{align*}
  8. \begin{align*}-5=\sqrt[5]{x-5}-7\end{align*}
  9. \begin{align*}\sqrt[4]{x-6}+10=13\end{align*}
  10. \begin{align*}\frac{8}{5}\sqrt[3]{x+5}=8\end{align*}
  11. \begin{align*}3 \sqrt{x+7}-2=25\end{align*}
  12. \begin{align*}\sqrt[4]{235+x}+9=14\end{align*}

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 7.7. 

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Vocabulary

TermDefinition
Extraneous Solution An extraneous solution is a solution of a simplified version of an original equation that, when checked in the original equation, is not actually a solution.
Quadratic Equation A quadratic equation is an equation that can be written in the form =ax^2 + bx + c = 0, where a, b, and c are real constants and a\ne 0.
Quadratic Formula The quadratic formula states that for any quadratic equation in the form ax^2+bx+c=0, x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}.
Radical Expression A radical expression is an expression with numbers, operations and radicals in it.

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