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Solve Radical Equation: Square Root of w^2 + 7w - 39 equals Square Root of 10w + 1
Published on June 18, 2026
Algebra
This problem requires solving a radical equation by squaring both sides to eliminate the square roots, leading to a quadratic equation. We then solve the quadratic and check for extraneous solutions to find the valid value of w.
Question
Answer
To find the solution to the equation
1. Square both sides of the equation to eliminate the square roots:
2. Rearrange the equation into a standard quadratic form (
Subtract
Subtract
3. Factor the quadratic equation:
We need to find two numbers that multiply to
The factors of
We need to find two numbers that multiply to
The factors of
-
(sum: ) -
(sum: ) -
(sum: ) -
(sum: ) -
(sum: ) -
(sum: ) -
(sum: ) This is our pair.
So, the factored form is:
4. Solve for
Set each factor to zero:
Set each factor to zero:
5. Check for extraneous solutions:
We must ensure that the values under the square roots are non-negative.
We must ensure that the values under the square roots are non-negative.
-
For
: -
(Valid) -
(Valid) -
, so is a solution.
-
-
For
: -
(Not a real number) - Since we cannot take the square root of a negative number in the real number system,
is an extraneous solution.
-
Looking at the provided options:
The only valid solution is
Final Answer:
The solution is
The solution is