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Answer:

\[7 \div \left(\frac{3\sqrt{2}}{4}\right) = n^2\]

Now, let's solve for \(n\) using the quadratic formula:

The quadratic formula is: \[n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

In this equation:

- \(a\) is the coefficient of \(n^2\), which is 1 (since it's implied).

- \(b\) is the coefficient of \(n\), which is 0 (since there's no \(n\) term).

- \(c\) is the constant term on the right side, which is \(\frac{7}{3\sqrt{2}/4}\).

Now, we can substitute these values into the formula:

\[n = \frac{-(0) \pm \sqrt{(0)^2 - 4(1)\left(\frac{7}{3\sqrt{2}/4}\right)}}{2(1)}\]

Simplify further:

\[n = \pm \sqrt{-4\left(\frac{7}{3\sqrt{2}/4}\right)}\]

Now, calculate the value inside the square root:

\[n = \pm \sqrt{-4\left(\frac{7}{\frac{3\sqrt{2}}{4}}\right)}\]

To proceed with this calculation, you would simplify the expression inside the square root further. First, divide 7 by \(\frac{3\sqrt{2}}{4}\), and then take the square root. This will give you the approximate values of \(n\).

Answer:

Step-by-step explanation:

To solve the equation \(7 \div 3 \frac{\sqrt{2}}{4} = n^{2}\), we first simplify the left-hand side:

\(7 \div 3 \frac{\sqrt{2}}{4}\)

To divide by a fraction, we can multiply by its reciprocal:

\(7 \times \frac{4}{3 \sqrt{2}}\)

Next, we multiply the numerator and denominator by the conjugate of the denominator to rationalize the denominator:

\(7 \times \frac{4}{3 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{28\sqrt{2}}{3 \times 2} = \frac{14\sqrt{2}}{3}\)

Now, the equation can be rewritten as:

\(\frac{14\sqrt{2}}{3} = n^{2}\)

To solve for \(n\), we can take the square root of both sides of the equation:

\(\sqrt{\frac{14\sqrt{2}}{3}} = \sqrt{n^{2}}\)

Simplifying the left-hand side:

\(\sqrt{\frac{14\sqrt{2}}{3}} = \frac{\sqrt{14\sqrt{2}}}{\sqrt{3}}\)

Therefore, the solution to the equation using the quadratic formula is:

\(n = \frac{\sqrt{14\sqrt{2}}}{\sqrt{3}}\)