To determine the solutions of the given quadratic equation [tex]\((4y - 3)^2 = 72\)[/tex], we can follow these steps:
1. Write down the given equation:
[tex]\[(4y - 3)^2 = 72\][/tex]
2. Take the square root of both sides of the equation to eliminate the square:
[tex]\[
4y - 3 = \pm \sqrt{72}
\][/tex]
3. Simplify [tex]\(\sqrt{72}\)[/tex]:
[tex]\[
\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}
\][/tex]
Therefore, we can write:
[tex]\[
4y - 3 = \pm 6\sqrt{2}
\][/tex]
4. Separate this into two distinct equations:
[tex]\[
4y - 3 = 6\sqrt{2}
\][/tex]
and
[tex]\[
4y - 3 = -6\sqrt{2}
\][/tex]
5. Solve each equation for [tex]\(y\)[/tex]:
- For [tex]\(4y - 3 = 6\sqrt{2}\)[/tex]:
[tex]\[
4y = 6\sqrt{2} + 3
\][/tex]
[tex]\[
y = \frac{6\sqrt{2} + 3}{4}
\][/tex]
- For [tex]\(4y - 3 = -6\sqrt{2}\)[/tex]:
[tex]\[
4y = -6\sqrt{2} + 3
\][/tex]
[tex]\[
y = \frac{3 - 6\sqrt{2}}{4}
\][/tex]
6. Hence, the solutions to the quadratic equation are:
[tex]\[
y = \frac{3 + 6\sqrt{2}}{4} \quad \text{and} \quad y = \frac{3 - 6\sqrt{2}}{4}
\][/tex]
Given the results, the correct option is:
[tex]\[
\boxed{y = \frac{3+6\sqrt{2}}{4} \text{ and } y = \frac{3-6\sqrt{2}}{4}}
\][/tex]
