EQUATIONS CONTAINING VARIABLES UNDER ONE OR MORE RADICALS
Recall the following:
Solve for x in the following equation.
Problem 2.5a:
Answer: x=1.
Solution
First make a note of the fact that you cannot take the square root of a
negative number. Therefore,the term is valid only if
, the term
is valid if , and the term is valid only if
. The equation is
valid if all three terms are valid, therefore the domain is restricted
to the common domain of the three terms or the set of real numbers .
Square both sides of the equation and simplify.
Isolate the
term and simplify.
Square both sides of the equation and simplify.
Use the quadratic formula to solve for x.
The answers are x=1, -0.017995.
Check the solution by substituting 1 in the original equation for x. If the
left side of the equation equals the right side of the equation after the
substitution, you have found the correct answer.
Check the solution by substituting -0.017995 in the original
equation for x. If the left side of the equation equals the right side of the equation
after the substitution, you have found the correct answer.
Since the left side of the original equation does not equal the right side of the original equation after we substituted -0.017995 for x, then the solution x=-0.017995 is a not valid and not a solution after all.
You can also check the answer by graphing the equation:
The graph represents the right side of the original equation minus the left side of the original equation. The x-intercept(s) of this graph is(are) the solution(s). Since there is just one x-intercept at 1, then the only solution is x=1.
If you would like to review the solution to problem 2.5b, click on solution.
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Author:Nancy Marcus