Root equations

Functions: Power functions and root functions

Root equations

A root equation is an equation involving root functions. In general, we can solve a root equation with the following #4# steps.

Solving root equations

	Procedure We solve a root equation for #x#.	Example #\sqrt{x+4}+4=9#
Step 1	Isolate the root. This means that by means of reduction, we make sure the root is the only thing on one side of the equation.	#\sqrt{x+4}=5#
Step 2	Take the square on both sides to get rid of the root.	#x+4=25#
Step 3	Solve this equation.	#x=21#
Step 4	Check if the found solution is a solution to the original equation.	#\sqrt{21+4}+4=9# Hence, the solution is correct.

CheckingChecking the solution(s) is necessary, because squaring something can result in solutions that are not solutions to the original equation. This can be seen in the second example at the top of this page.

Solve the equation below:
\[\sqrt{6\cdot x-2}=\sqrt{5-3\cdot x}\]
Give your answer in the form of

#none# if there is no solution,
#x=x_1# if there is one solution,
#x=x_1\lor x=x_2# if there are two solutions,

with the correct values for #x_1# and #x_2#.

# x={{7}\over{9}} #

\[\begin{array}{rcl}
\sqrt{6\cdot x-2}&=& \sqrt{5-3\cdot x}\\
&&\phantom{xxx}\blue{\text{original equation}}\\

6\cdot x-2&=&5-3\cdot x \\
&&\phantom{xxx}\blue{\text{both sides squared}} \\
9\cdot x&=&7 \\
&&\phantom{xxx}\blue{\text{terms of }x \text{ brought to the left, constant terms brought to the right}} \\

x&=&{{7}\over{9}} \\
&&\phantom{xxx}\blue{\text{divided by the coefficient of }x} \\
\end{array}\]

Because we have taken the square, the solution for #x# that we found may not be a solution to the original equation. Therefore, we must now test the solution we found by entering it into the original equation.

On the left side is:
\[\sqrt{6\cdot \left({{7}\over{9}}\right)-2}={{2^{{{3}\over{2}}}}\over{\sqrt{3}}}\]
On the right side is:
\[\sqrt{5-3\cdot \left({{7}\over{9}}\right)}={{2^{{{3}\over{2}}}}\over{\sqrt{3}}}\]
Left and right are equal, so this solution is correct.

In conclusion, the answer to the equation is # x={{7}\over{9}} #.

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