How to Solve Radical Equations

Step-by-Step Guide

1. 1

   Step 1: Isolate the variable and radical on one side of the equation.

   This is just like solving for any other algebraic equation.
   
   Combine like terms and add/subtract numbers so that your variable and radical stand alone.
   
   If it helps, treat the x{\displaystyle {\sqrt {x}}} like a normal "x" in any other problem, and solve for that.
   
   For example, with the problem x+3=10{\displaystyle {\sqrt {x}}+3=10}:
   Isolate x{\displaystyle {\sqrt {x}}}: x+3=10{\displaystyle {\sqrt {x}}+3=10} Subtract 3 from both sides: x+3−3=10−3{\displaystyle {\sqrt {x}}+3-3=10-3} Simplify both sides: x=7{\displaystyle {\sqrt {x}}=7}
2. 2

   Step 2: Square both sides of the equation to remove the radical.

   All you have to do to undo a radical is square it.
   
   Because you need the equation to stay balanced, you square both sides, just like you added or subtracted from both sides earlier.
   
   So, for the example:
   Isolate x{\displaystyle {\sqrt {x}}}: x=7{\displaystyle {\sqrt {x}}=7} Square both sides: (x)2=(7)2{\displaystyle ({\sqrt {x}})^{2}=(7)^{2}} Final answer: x=49{\displaystyle x=49}, When solving radical equations, you can get answers that don't actually fit the problem.
   
   You must always check your solutions to make sure you have all real answers.
   
   To check an answer, simply plug in each answer for "x" in the original equation:
   Original Equation: x+3=10{\displaystyle {\sqrt {x}}+3=10} Substitute 49 for x: 49+3=10{\displaystyle {\sqrt {49}}+3=10} Solve: 7+3=10{\displaystyle 7+3=10} Our solution is valid: 10=10{\displaystyle 10=10}, This same strategy works no matter what the root, like x3−1=3{\displaystyle {\sqrt{x}}-1=3}.
   
   You just need to raise both sides to the same power as the root.
   
   So, for this example:
   Isolate x3{\displaystyle {\sqrt{x}}}: x3−1=3{\displaystyle {\sqrt{x}}-1=3} Add 1 to both sides: x3−1+1=3+1{\displaystyle {\sqrt{x}}-1+1=3+1} Simplify both sides: x3=4{\displaystyle {\sqrt{x}}=4} Cube both sides: (x3)3=(4)3{\displaystyle ({\sqrt{x}})^{3}=(4)^{3}} Final Answer: 64 Check Solution: 643−1=4−1=3{\displaystyle {\sqrt{64}}-1=4-1=3}, When removing the radical, you square both sides of the equation.
   
   If you have multiple terms, such as the equation x=2x+3{\displaystyle {\sqrt {x}}=2x+3}, you must square the entire side, not the individual terms (2x2{\displaystyle 2x^{2}} and 32{\displaystyle 3^{2}} are both incorrect).
   
   Before solving for x in the example, then:
   Original Equation: x=2x+3{\displaystyle {\sqrt {x}}=2x+3} Square both sides: (x)2=(2x+3)2{\displaystyle ({\sqrt {x}})^{2}=(2x+3)^{2}} Expand Expressions: x=4x2+12x+9{\displaystyle x=4x^{2}+12x+9} The expression above was expanded through Polynomial Multiplication.
   
   If you're confused how it was done, you can review the process here.
3. 3

   Step 3: Check your answers in the original problem.

4. 4

   Step 4: Use the same technique for more complicated roots

5. 5

   Step 5: not just squares.

6. 6

   Step 6: Remember to square both sides of the equation

7. 7

   Step 7: not just the terms.

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