Solving Radical Equations with Two Radical Terms

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  • 0:04 Radical Equations
  • 1:12 Two Radical Terms
  • 5:30 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

Solving equations with two radical terms takes some patience and care, but it really is not difficult. This lesson will show you the steps to solve these more complicated equations plus give you some examples to follow.

Radical Equations

A radical equation is an equation containing a radical, or square root symbol. Here is an example of a radical equation:


The method used to solve radical equations starts by isolating the radical, then squaring both sides of the equation. By squaring the radical, we can remove the square root from the equation, making it easier to solve. When the radical has been removed, we can easily solve for x.

An often forgotten step when solving radical equations is to check your answer. Radicals are funny things, and not every solution is a true solution. It's easy to breathe a sigh of relief when we see that 'x = ' and move on to the next problem. But it is very important to check your answer when working with square roots.

The way to do this is to take the answer you've calculated and substitute it back into the original problem. Substitute 13 for x and solve. Since the square root of 16 is equal to 4, we know that we got a true answer.

Radical Equations with Two Radical Terms

What if the equation has more than one radical term? If that is the only thing it has, two terms, each under a radical sign, then you just need to square both of them and then solve.

sqrt(5x + 3) = sqrt(3x + 7)

Square both sides. The square and square root cancel each other out, leaving 5x + 3 = 3x + 7. Then, solve for x. 2x = 4 and x = 2.

What if the equation has other terms besides the two radicals? It makes the solution a bit more difficult to solve but certainly not impossible. We just need to perform the steps used to solve a radical equation with one radical twice. Here's an example. Solve √ (x - 3) + √x = 3.

The first step is always to isolate the radical symbols on one side of the equation. Since this problem already has the radicals only on the left, we can move on to the next step. The next step is to square both sides of the equation.

Remember, when you square a binomial, you use the FOIL method to multiply the binomial to itself. The FOIL method is a simple way to remember what terms need to be multiplied together. It's a mnemonic, which stands for:

F = First - multiply the first two terms in each binomial
O = Outside - multiply the outside terms in each binomial
I = Inside - multiply the inside terms in each binomial
L = Last - multiply the last two terms in each binomial

When we write out the left side of this equation in order to solve it, it looks like this:


When we multiply the first two terms, we get x - 3 because √(x - 3) times itself is just x - 3. Multiplying the outside terms is a bit trickier. It looks like this:


We need to distribute the square root of x, multiply it by both the x and the 3 in the other term. The answer we get is this:


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