A radical equation contains at least one unknown beneath a radical symbol -- often a square root. Some equations that contain multiple radicals may require more steps, but the basic techniques for solving all radical equations is the same.

Solve a Basic Equation

The simplest square root equation consists of a radical on one side of the equal sign and a value on the other, as shown below:

sqrt(x) = 5

Solve for x by squaring both sides of the equation to get the following:

x = 5^2

X's value in this example is 25.

Radical Equations with Multiple Terms

You'll find more complex equations that contain several terms on the radical side of the equation, as seen below:

sqrt(x) + 5 = 17

Before you square both sides of the equation, isolate the radical by subtracting 5 from both sides of the equation to obtain sqrt(x) = 17-5. Square both sides of the equation, and you get the following:

x = 12^2 x = 144

Begin Solving a Two Square Root Problem

When an equation contains two radicals, the math gets a little trickier. Suppose you have this equation:

sqrt(x - 3) + sqrt(x) = 10

Isolate one of the radicals by shifting other terms to the other side of the equation, as seen below:

sqrt(x - 3) = 10 - sqrt(x)

Square both sides to obtain this equation:

x - 3 = (10 - sqrt(x))^2

That's the same as this expanded equation:

x - 3 = (10 - sqrt(x)) * (10 - sqrt(x))

Finish Solving a Two Square Root Problem

Continuing from your previous efforts in solving a radical equation with two square roots, you multiply terms on the right side of the equation and simplify them further to get the following:

x - 3 = (10*10) - (10 * sqrt(x)) - (10 * sqrt(x)) + x x - 3 = 100 - 10 * sqrt(x) - 10 * sqrt(x) + x x - 3 = 100 - 20 * sqrt(x) + x

Simplify the final equation by subtracting x from both sides and adding 3 to both sides to yield these equations:

0 = 100 - 20 * sqrt(x) + 3 0 = 103 - 20 * sqrt(x) 20 * sqrt(x) = 103 sqrt(x) = 103/20 sqrt(x) = 5.15

Square both sides to get x = 26.52

Validate The Answer

Always verify that your solution is correct by plugging it back into the original equation. Consider the previous example that has the following equation:

sqrt (x - 3) + sqrt(x) = 10

Replace x with the answer, 26.52, and the equation appears as shown below:

sqrt(26.52 - 3) + sqrt( 26.52) = 10

Solve the equation to verify that the answer is correct