
How do you find extraneous solutions from radical equations?
Answer
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Hint: An extraneous solution is a solution value of the variable in the equations, that is found by solving the given equation algebraically but it is not a solution of the given equation. We get these types of solutions sometimes when we solve radical equations. Radical equations are equations in which the variable is under a radical. To find the solution of a radical equation, we need to take the radical expression to one side of the equations. If there is more than one radical expression, take one at a time.
Complete step by step answer:
Let’s take an example of radical expression . We need to solve this equation. Squaring both sides of the above equation, we get
simplifying the above equation, we get
We can find the roots of the above quadratic equation, by using the formula method, as follows
or
or
But if we substitute in the equation, we get
Which is not correct. Hence, is not a solution of the given radical equation. Thus is an extraneous solution for the given equation.
Note: We should know when an extraneous solution occurs. Extraneous solutions of an equation are solutions that occur when a radical expression that has an even index, such as 2, is raised to its power to find the solution of an equation.
In the above example, as the radical power has an even index, we get as an extraneous solution of the equation.
Complete step by step answer:
Let’s take an example of radical expression
simplifying the above equation, we get
We can find the roots of the above quadratic equation, by using the formula method, as follows
But if we substitute
Which is not correct. Hence,
Note: We should know when an extraneous solution occurs. Extraneous solutions of an equation are solutions that occur when a radical expression that has an even index, such as 2, is raised to its power to find the solution of an equation.
In the above example, as the radical power has an even index, we get
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