Let's solve the equation [tex]\(\sqrt{a + 4} + \sqrt{a - 1} = 5\)[/tex] step by step.
1. Isolate one of the square roots on one side of the equation:
We can isolate [tex]\(\sqrt{a + 4}\)[/tex]:
[tex]\[
\sqrt{a + 4} = 5 - \sqrt{a - 1}
\][/tex]
2. Square both sides of the equation:
Squaring both sides will help us eliminate the square root on the left-hand side:
[tex]\[
(\sqrt{a + 4})^2 = (5 - \sqrt{a - 1})^2
\][/tex]
This yields:
[tex]\[
a + 4 = 25 - 10\sqrt{a - 1} + (a - 1)
\][/tex]
3. Simplify the equation:
Combine like terms on the right-hand side:
[tex]\[
a + 4 = 24 + a - 10\sqrt{a - 1}
\][/tex]
4. Isolate the term with the square root:
Subtract [tex]\(a\)[/tex] and 24 from both sides:
[tex]\[
4 - 24 = -10\sqrt{a - 1}
\][/tex]
[tex]\[
-20 = -10\sqrt{a - 1}
\][/tex]
5. Divide both sides by -10:
[tex]\[
\sqrt{a - 1} = 2
\][/tex]
6. Square both sides again:
[tex]\[
(\sqrt{a - 1})^2 = 2^2
\][/tex]
[tex]\[
a - 1 = 4
\][/tex]
7. Solve for [tex]\(a\)[/tex]:
Add 1 to both sides:
[tex]\[
a = 5
\][/tex]
We've found that [tex]\(a = 5\)[/tex].
8. Verify the solution:
Substitute [tex]\(a = 5\)[/tex] back into the original equation to ensure it satisfies:
[tex]\[
\sqrt{5 + 4} + \sqrt{5 - 1} = \sqrt{9} + \sqrt{4} = 3 + 2 = 5
\][/tex]
Since both sides are equal, the solution [tex]\(a = 5\)[/tex] is correct. Therefore, the value of [tex]\(a\)[/tex] that satisfies the equation [tex]\(\sqrt{a + 4} + \sqrt{a - 1} = 5\)[/tex] is [tex]\(\boxed{5}\)[/tex].
