Section 6.6
Radical Equations
A radical equation is an equation that includes a radical (normally, a square root), and that has the variable inside the radical. Typical examples are $\sqrt{2x+3}-7=4$ and \[x+\sqrt{11-x}=5\]. More complicated examples are possible, such as $1+\sqrt[3]{x-4}=3$.
The first step in solving a radical equation is to isolate the radical. That is to say, get the radical by itself on one side of the equation.
The second step is to square (or cube) both sides of the equation in order to eliminate the radical.
Step three is to solve the resulting equation for the variable.
The last step is to check your answers in the original equation! Squaring both sides of an equation can introduce what are known as "extraneous solutions"—numbers that are not solutions of the original equation.
Examples:
\begin{align}
\text{a. }\sqrt{2x+3}-7&=4 \\
\sqrt{2x+3}&=11 \\
2x+3&=121 \\
2x&=118 \\
x&=59
\end{align}
Check:
\begin{align}
\sqrt{2\cdot 59+3}-7&=\sqrt{118+3}-7 \\
& =\sqrt{121}-7 \\
& =11-7 \\
& =4
\end{align}
The solution checks.
\begin{align}
\text{b. }x+\sqrt{11-x}&=5 \\
\sqrt{11-x}&=5-x \\
11-x&=25-10x+{{x}^{2}} \\
0&={{x}^{2}}-9x+14 \\
0&=(x-7)(x-2) \\
x&=2,\; 7
\end{align}
Check the solutions:
If $x=2$, then we have $2+\sqrt{11-2}=2+\sqrt{9}=2+3=5$. This solution works.
If $x=7$, then we have $7+\sqrt{11-7}=7+\sqrt{4}=7+2=9$. This solution does not check.
The only solution of the original equation is $x=2$.
Solving radical equations: Solving Radical Equations
Extraneous solutions to radical equations: Extraneous Solutions to Radical Equations
Exercises:
Solve the following equations for $x$.
\begin{array}{l l l}
1. \;\;\;\sqrt{x+3}+2=5\;\;\; & 2. \;\;\;\sqrt{x-3}-4=1\;\;\; & 3. \;\;\;3-\sqrt{x-2}=1\;\;\; \\
4. \;\;\;4-\sqrt{10-x}=2\;\;\; & 5. \;\;\;x+\sqrt{x-3}=5\;\;\; & 6. \;\;\;\sqrt{x+1}+x=11\;\;\; \\
7. \;\;\;\sqrt{x}-x=0\;\;\; & 8. \;\;\;\sqrt{x}+x=2\;\;\; & 9. \;\;\;\sqrt{16+3x}-x=2\;\;\; \\
10. \;\;\;\sqrt{16-2x}+x=10\;\;\; & 11. \;\;\;\sqrt[3]{x+5}+1=3\;\;\; & 12. \;\;\;\sqrt[3]{2x+15}-1=2\;\;\; \\
13. \;\;\;x-\sqrt{x}=2x-6\;\;\; & 14. \;\;\;x+\sqrt{3x+1}=2x+1\;\;\; & 15. \;\;\;2x-\sqrt{3x+1}=x+1\;\;\; \\
16. \;\;\;4x-\sqrt{5x+1}=3x-1\;\;\; & 17. \;\;\;2\sqrt{x+1}+1=x+2\;\;\; & 18. \;\;\;10-2\sqrt{x+1}=x-2\;\;\; \\
19. \;\;\;x+3\sqrt{x-1}=2x+1\;\;\; & 20. \;\;\;\frac{x}{5}+\frac{\sqrt{x+4}}{3}=2\;\;\; & 21. \;\;\;3-\sqrt{2x+5}=-x\;\;\;
\end{array}