Pronunciation: /səˈlu.ʃən/ Explain

The solution to a math problem is the correct answer for the problem. The solution to an equation is a set of one or more values that, when substituted for a variable, make an equation or system true or consistent. To solve a math problem is to find the solution of the problem.

For example, given the equation x + 3 = 5, the solution is x = 2. x = 2 satisfies the equation x + 3 = 5. Try substituting 2 for x in the equation. The equation becomes 2 + 3 = 5. Since this is a true statement, we know that 2 is a solution to the equation.

An extraneous solution is a apparent solution that does not satisfies the equation or system. Extraneous solutions can occur when multiplying both sides of an equation by a variable, and by squaring both sides of an equation:

How to Find an Extraneous Solution
1x-2=square root(x). This is the equation to solve.
2(x-2)^2=(square root(x))^2. Square both sides of the equation. This is the step where and extraneous solution is introduced. Expand both sides of the equation by applying the exponents.
3x-4x+4=x implies x-5x+4=0. Subtract x from both sides of the equation.
4x-5x+4=0 implies (x-4)(x-1)=0. Factor the left hand side of the equation.
5(x-4)(x-1)=0 implies x=4 or x=1. The factors imply solutions of 4 and 1.
6x-2=square root(x), x=4 implies 4-2=square root(2) implies 2=2. Check the solution x = 4 by substituting 4 for x in the original equation. The equation now simplifies to 2 = 2. Since this is a true statement that satisfies the equation, 2 is a solution to the equation. 2 is not an extraneous solution to the equation.
7x-2=square root(x), x=1 implies 1-2=square root(1) implies -1=1. Check the solution x = 1 by substituting 1 for x in the original equation. The equation now simplifies to -1 = 1. Since this is a false statement, 1 does not satisfy the equation. 1 is a not solution to the equation. 1 is an extraneous solution.
Table 1: Finding an extraneous solution.


  1. McAdams, David E.. All Math Words Dictionary, solution. 2nd Classroom edition 20150108-4799968. pg 168. Life is a Story Problem LLC. January 8, 2015. Buy the book

Cite this article as:

McAdams, David E. Solution. 5/7/2019. All Math Words Encyclopedia. Life is a Story Problem LLC.

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Revision History

5/7/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
12/7/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
1/9/2009: Added section on 'extraneous solution'. (McAdams, David E.)
12/3/2008: Added 'one or more values' to the definition. (McAdams, David E.)
9/15/2008: Added 'More Information'. (McAdams, David E.)
3/5/2008: Add solution to a math problem. (McAdams, David E.)
7/11/2007: Initial version. (McAdams, David E.)

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