Definition--Rationals and Radicals--Extraneous Solution
Solving equations is an important skill, but sometimes you end up finding more solutions than you realize. Watch the following video to learn how to identify extraneous solutions when they occur. (The transcript is also included.)
The following section provides additional information about extraneous solutions. Use it as a review resource before or after watching the video.
A Review of Equations
Suppose you want to solve this equation:
x + 2 = 5
As written, this is known as a conditional equation. That means that if you plug in random values for x, the equation is either true or false. Plug in x = 1, 2, and 3 into the equation.
|x = 1||1 + 2 = 5||False|
|x = 2||2 + 2 = 5||False|
|x = 3||3 + 2 = 5||True|
Only one value (x = 3) makes the equation true. The other values, as well as any other values for x, make the equation false. The value x = 3 is the solution to the equation. Some equations have more than one solution. For example, quadratic equations and polynomial equations often have more than one solution.
What Are Extraneous Solutions?
The video was uploaded on 9/7/2022.
You can view the video here.
The video lasts for 2 minutes and 50 seconds.
When you solve an equation, what you are doing is finding the value or values of x that make an equation true.
Before solving it, the equation is what's called a conditional equation, which is another way of asking, for what values of x is this equation true?
Here's an example.
The equation 2x + 3 = 17 is a conditional equation.
The equation is true for only one value of x.
Solving the equation is finding the value of x that makes the equation true.
Here is the solution, x = 7.
When you solve the equation and you find the correct value for x, you can follow one more step.
Go back to the original equation and plug in the value you found for x.
Simply plug-in 7 into the original equation and simplify.
As you can see, when x = 7, the conditional equation becomes a true equation.
Yet, there are times when solving an equation results in more solutions than expected.
In these cases, one of the solutions may not result in a true equation.
Let's look at an example.
Here is an equation.
The square root of the quantity x + 4 equals x minus 8.
To solve for x, first square both sides of the equation.
The term on the left becomes x + 4.
The term on the right expands into a quadratic expression.
Now combine all terms to one side of the equation, as shown.
The result is a quadratic equation.
It can be factored into the product of two binomials, as shown.
There are two solutions, x equals 5 and x equals 12.
Let's circle back and plug-in these solutions into the original equation.
For x = 12, the solution results in a true equation.
However, for x = 5, the solution does not result in a true equation.
This means that x = 5 is an extraneous solution.
Here's another way of looking at this equation.
One way to solve equations is to treat each side of the equation as a separate function.
Graph each function and find where they intersect.
The intersection is the solution.
As you can see, the graphs intersect at only one point, where x = 12.
So, where does the extraneous solution come from?
Going back to the original equation, the square root term also has a negative root, as shown here.
By convention, with square roots we use the positive root, but the graph of the negative root reveals the extraneous solution.
With this graph, the extraneous solution is revealed for x = 5.
A Review of Graphical Solutions
Suppose you want to solve this equation:
This equation is hard to solve algebraically. We can solve this equation graphically by finding where these functions intersect:
Here is what the graphs of those functions look like. Where they intersect is the solution to the equation.
The graphs intersect at the point (8, 3). This means that the solution to this equation is x = 8.
Here’s another way to solve the same equation:
Notice that this version has two solutions: x = 3 and x - 8. Notice that one of the solutions (x = 8) is the same as the solution we found by graphing. But what about the other solution, x = 3? Where did this solution come from and why didn’t we find it by graphing? Why does x = 3 not satisfy the original equation?
The solution x = 3 is called an extraneous solution. It looks like a solution, but it isn’t. You can plug x = 3 into the original equation and you’ll see that it results in a false equation.
By convention, the square root of a number is the positive root. Therefore, x = 3 is not a solution to this equation because it results in a false equation.
Where Extra Solution Comes From
Take a look at the original graphical solution.
Now look at this graphical solution. This one has the negative root for the first function.
This version has a solution of x = 3, but it’s based on the negative root of one of the functions. Since this root is normally discarded, this is the reason this is an extraneous solution.
You can see that a graphical approach is a way to filter out extraneous solutions. The algebraic solution can help you find the extraneous solution.
Here’s another example.
Multiply both sides by x2 - 9 and simplify.
According to the algebraic solution, x = 3 is a solution. But it can’t be a solution, because in the original equation it would result in a denominator of zero.
This is reflected in the graphical solution. Notice that the two functions don’t intersect.
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|Common Core Standards||CCSS.MATH.CONTENT.HSA.REI.A.2, CCSS.MATH.CONTENT.HSN.RN.A.1, CCSS.MATH.CONTENT.HSF.IF.C.7|
|Grade Range||8 - 12|
• Rational Expressions and Functions
• Rational Functions and Equations
|Keywords||radicals, radical expressions, rational numbers, rational expressions, definitions, glossary term, rational functions|