To solve the equation [tex]\(3 \sqrt{x} = \sqrt{4x + 3}\)[/tex], let’s follow these steps:
1. Square both sides of the equation to eliminate the square roots. By doing this, we can work with a simpler polynomial form.
[tex]\[
(3 \sqrt{x})^2 = (\sqrt{4x + 3})^2
\][/tex]
2. Simplify both sides:
[tex]\[
9x = 4x + 3
\][/tex]
3. Move all terms involving [tex]\(x\)[/tex] to one side of the equation and constants to the other side. Subtract [tex]\(4x\)[/tex] from both sides:
[tex]\[
9x - 4x = 3
\][/tex]
[tex]\[
5x = 3
\][/tex]
4. Solve for [tex]\(x\)[/tex] by dividing both sides by 5:
[tex]\[
x = \frac{3}{5}
\][/tex]
5. Verify the solution by substituting [tex]\(x = \frac{3}{5}\)[/tex] back into the original equation:
[tex]\[
3 \sqrt{\frac{3}{5}} = \sqrt{4 \left(\frac{3}{5}\right) + 3}
\][/tex]
Calculate inside the square roots:
[tex]\[
3 \sqrt{\frac{3}{5}} \quad \text{and} \quad \sqrt{4 \cdot \frac{3}{5} + 3}
\][/tex]
[tex]\[
3 \sqrt{\frac{3}{5}} = 3 \cdot \sqrt{\frac{3}{5}}
\][/tex]
[tex]\[
\sqrt{4 \cdot \frac{3}{5} + 3} = \sqrt{\frac{12}{5} + 3} = \sqrt{\frac{12}{5} + \frac{15}{5}} = \sqrt{\frac{27}{5}} = \sqrt{\frac{27}{5}}
\][/tex]
After taking the square root of both sides and simplifying we get:
[tex]\[
3 \sqrt{\frac{3}{5}} = \sqrt{4 \left(\frac{3}{5}\right) + 3}
\][/tex]
Both sides match, confirming our solution is correct.
Therefore, the solution to the equation [tex]\(3 \sqrt{x} = \sqrt{4x + 3}\)[/tex] is:
[tex]\[
x = \frac{3}{5}
\][/tex]
