To solve the equation [tex]\(\frac{a-3}{\sqrt{a}} = -\sqrt{a}\)[/tex], we will proceed through a series of steps to simplify and solve for [tex]\(a\)[/tex].
### Step 1: Clear the Fraction
First, we want to eliminate the fraction by multiplying every term by [tex]\(\sqrt{a}\)[/tex]. This results in:
[tex]\[
\frac{a-3}{\sqrt{a}} \cdot \sqrt{a} = -\sqrt{a} \cdot \sqrt{a}
\][/tex]
### Step 2: Simplify
Simplifying the left and right-hand sides, we get:
[tex]\[
a - 3 = -a
\][/tex]
### Step 3: Solve for [tex]\(a\)[/tex]
To isolate [tex]\(a\)[/tex], add [tex]\(a\)[/tex] to both sides of the equation:
[tex]\[
a - 3 + a = -a + a
\][/tex]
[tex]\[
2a - 3 = 0
\][/tex]
Next, add 3 to both sides:
[tex]\[
2a - 3 + 3 = 3
\][/tex]
[tex]\[
2a = 3
\][/tex]
Now, divide by 2:
[tex]\[
a = \frac{3}{2}
\][/tex]
### Step 4: Verify the Solution
We should verify that [tex]\(a = \frac{3}{2}\)[/tex] satisfies the original equation. Substituting [tex]\(a = \frac{3}{2}\)[/tex] back in:
[tex]\[
\frac{\frac{3}{2} - 3}{\sqrt{\frac{3}{2}}} = -\sqrt{\frac{3}{2}}
\][/tex]
Calculate [tex]\(\frac{3}{2} - 3\)[/tex]:
[tex]\[
\frac{3}{2} - \frac{6}{2} = -\frac{3}{2}
\][/tex]
Calculate [tex]\(\sqrt{\frac{3}{2}}\)[/tex]:
[tex]\[
\sqrt{\frac{3}{2}}
\][/tex]
So our equation simplifies to:
[tex]\[
\frac{-\frac{3}{2}}{\sqrt{\frac{3}{2}}} = -\sqrt{\frac{3}{2}}
\][/tex]
Simplify the left side:
[tex]\[
\frac{-\frac{3}{2}}{\sqrt{\frac{3}{2}}} = -\sqrt{\frac{3}{2}}
\][/tex]
This is true because:
[tex]\[
\frac{-3}{2} \div \sqrt{\frac{3}{2}} = -\sqrt{\frac{3}{2}}
\][/tex]
### Conclusion
Therefore, [tex]\(\boxed{\frac{3}{2}}\)[/tex] is indeed the solution to the equation [tex]\(\frac{a-3}{\sqrt{a}} = -\sqrt{a}\)[/tex].
