# What are common mistakes students make with respect to extraneous solutions?

Aug 5, 2017

A couple of thoughts...

#### Explanation:

These are more guesses than informed opinion, but I would suspect the main error is along the lines of not checking for extraneous solutions in the following two cases:

• When solving the original problem has involved squaring it somewhere along the line.

• When solving a rational equation and having multipled both sides by some factor (which happens to be zero for one of the roots of the derived equation).

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Example 1 - Squaring

Given:

$\sqrt{x + 3} = x - 3$

Square both sides to get:

$x + 3 = {x}^{2} - 6 x + 9$

Subtract $x + 3$ from both sides to get:

$0 = {x}^{2} - 7 x + 6 = \left(x - 1\right) \left(x - 6\right)$

Hence $x = 1$ or $x = 6 \text{ }$ (but $x = 1$ is not a valid solution of the original equation)

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Example 2 - Rational equation

Given:

${x}^{2} / \left(x - 1\right) = \frac{3 x - 2}{x - 1}$

Multiply both sides by $\left(x - 1\right)$ to get:

${x}^{2} = 3 x - 2$

Subtract $3 x - 2$ from both sides to get:

$0 = {x}^{2} - 3 x + 2 = \left(x - 1\right) \left(x - 2\right)$

Hence $x = 1$ or $x = 2 \text{ }$ (but $x = 1$ is not a valid solution of the original equation)