To solve the equation [tex]\(\sqrt{5x+3} = 3\sqrt{x}\)[/tex], let's follow a step-by-step approach.
First, we need to isolate the radical expressions by squaring both sides of the equation to eliminate the square roots.
The given equation is:
[tex]\[
\sqrt{5x + 3} = 3 \sqrt{x}
\][/tex]
Square both sides to remove the square roots:
[tex]\[
(\sqrt{5x + 3})^2 = (3 \sqrt{x})^2
\][/tex]
This simplifies to:
[tex]\[
5x + 3 = 9x
\][/tex]
Next, we need to solve for [tex]\(x\)[/tex]. Move all the terms involving [tex]\(x\)[/tex] to one side of the equation:
[tex]\[
5x + 3 - 9x = 0
\][/tex]
Combine the like terms:
[tex]\[
-4x + 3 = 0
\][/tex]
Now, isolate [tex]\(x\)[/tex] by moving the constant term to the other side:
[tex]\[
-4x = -3
\][/tex]
Divide both sides by [tex]\(-4\)[/tex]:
[tex]\[
x = \frac{3}{4}
\][/tex]
Finally, we should check if this solution is valid by substituting [tex]\(x = \frac{3}{4}\)[/tex] back into the original equation to ensure it satisfies the equation:
[tex]\[
\sqrt{5 \left(\frac{3}{4}\right) + 3} = 3 \sqrt{\frac{3}{4}}
\][/tex]
Calculate inside the square roots:
[tex]\[
\sqrt{\frac{15}{4} + 3} = 3 \cdot \sqrt{\frac{3}{4}}
\][/tex]
Convert 3 to a fraction with a common denominator:
[tex]\[
\sqrt{\frac{15}{4} + \frac{12}{4}} = 3 \cdot \sqrt{\frac{3}{4}}
\][/tex]
Simplify inside the square roots:
[tex]\[
\sqrt{\frac{27}{4}} = 3 \cdot \sqrt{\frac{3}{4}}
\][/tex]
Simplify further:
[tex]\[
\frac{\sqrt{27}}{2} = 3 \cdot \frac{\sqrt{3}}{2}
\][/tex]
Since [tex]\(\sqrt{27} = 3\sqrt{3}\)[/tex], the equation becomes:
[tex]\[
\frac{3\sqrt{3}}{2} = 3 \cdot \frac{\sqrt{3}}{2}
\][/tex]
Both sides are equal:
[tex]\[
\frac{3\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}
\][/tex]
Thus, the solution [tex]\(x = \frac{3}{4}\)[/tex] is verified to be correct.
Therefore, the solution to the equation [tex]\(\sqrt{5x+3} = 3\sqrt{x}\)[/tex] is:
A. [tex]\(\frac{3}{4}\)[/tex]
