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An extraneous solution is a solution that is derived from the original equation, but it is not actually a solution.
See solution.
Let's start by recalling the definitions of two important concepts — rational expression and excluded value.
Rational Expression | Excluded Values |
---|---|
2/x-2 | x=2 |
y/y^2+3 | None |
2z+5/z^2-1 | z=- 1 and z=1 |
Now we need to consider two more definitions — rational equation and extraneous solution.
When we end up with an extraneous solution in a rational equation, it is because one of our solutions is an excluded value of one of the rational expressions in the original equation.
Rational Equation | Extraneous Solutions | Rational Expressions | Excluded Values |
---|---|---|---|
x^2-3x/x-1=- 2/x-1 | x=1 | x^2-3x/x-1 and - 2/x-1 | x=1 |
2y/y^2-1=y^2/y+1 | y=- 1 | 2y/y^2-1=y^2/y+1 | y=- 1 and y=1 |
z+3/z^2-4z+4=z-1/z-2+4 | None | z+3/z^2-4z+4 and z-1/z-2 | z=2 |
Consider the following rational equation. 4x+8/x^2+4x+3=3/x+1 We will solve it using two methods and compare our results.
Cross multiply
a^2-b^2=(a+b)(a-b)
Use the Zero Product Property
(I): LHS-1=RHS-1
(II): LHS+1=RHS+1
Substitute | Simplify | Extraneous Solution? |
---|---|---|
4( - 1)+8/( - 1)^2+4( - 1)+3? =3/- 1+1 | 4/0? =3/0 * | Yes |
4( 1)+8/1^2+4( 1)+3? =3/1+1 | 12/8=3/2 âś“ | No |
We can see that only x=1 satisfies the original equation and that x=- 1 is an extraneous solution.
a/b=a * (x+3)/b * (x+3)
LHS * (x+1)(x+3)=RHS* (x+1)(x+3)
The above example shows that the extraneous solutions depend on the method used. When we solved the equation using the Cross Products Property we obtained one valid solution and one extraneous solution. When we used the LCD, we got only the valid solution.