Question 7

The value of a is √2 + 1.

The equation given simplifies to show that the value of a must be √2 + 1 to hold true under the provided conditions.

A) 3

Option A is incorrect because substituting a = 3 into any derived expression from the original equation does not satisfy the equality of √24 + √12 = 2√3. The value 3 does not correlate with the simplifications of the radicals presented.

B) 6

Option B is also incorrect. A value of a = 6 does not align with the results of combining the radicals from the left side of the equation, as it fails to maintain the relationship necessary to equate to 2√3.

C) √3

Option C is incorrect because substituting a = √3 does not yield a correct result in the context of the equation provided. The left-hand side does not equal the right-hand side when √3 is used as a value for a.

D) 2√3

Option D is incorrect as well. If a were equal to 2√3, it would not satisfy the equation √24 + √12 = 2√3, as the left side does not resolve to the necessary equality involved.

E) √2 + 1

Option E is correct because substituting a = √2 + 1 into the equation yields a valid equality. This value aligns with the simplifications derived from √24 and √12, confirming that it is the solution that satisfies the original equation.

Conclusion

The correct answer is definitively E) √2 + 1 because it is the only option that maintains the validity of the equation √24 + √12 = 2√3. All other options fail to provide a valid solution when substituted back into the context of the equation, demonstrating their inadequacy in satisfying the conditions laid out in the problem.