Solve this:
a^2-\frac{169}{10404a^2}=\frac{13}{52}

We are given the equation:

\[
a^2 - \frac{169}{10404a^2} = \frac{13}{52}
\]

### Step 1: Simplify the equation

First, simplify the right-hand side of the equation. The fraction \(\frac{13}{52}\) simplifies to:

\[
\frac{13}{52} = \frac{1}{4}
\]

So the equation becomes:

\[
a^2 - \frac{169}{10404a^2} = \frac{1}{4}
\]

### Step 2: Simplify the term \(\frac{169}{10404a^2}\)

Notice that \(\frac{169}{10404}\) can be simplified:

\[
\frac{169}{10404} = \frac{169}{169 \times 62} = \frac{1}{62}
\]

So the equation becomes:

\[
a^2 - \frac{1}{62a^2} = \frac{1}{4}
\]

### Step 3: Eliminate fractions

To eliminate the fractions, multiply the entire equation by \(62a^2\) to get rid of the denominator:

\[
62a^2 \cdot \left(a^2 - \frac{1}{62a^2}\right) = 62a^2 \cdot \frac{1}{4}
\]

Simplifying both sides:

\[
62a^4 - 1 = \frac{62a^2}{4}
\]

### Step 4: Multiply through by 4

Now, multiply through by 4 to eliminate the denominator on the right-hand side:

\[
4(62a^4 - 1) = 62a^2
\]

This simplifies to:

\[
248a^4 - 4 = 62a^2
\]

### Step 5: Rearrange the equation

Move all terms to one side:

\[
248a^4 - 62a^2 - 4 = 0
\]

### Step 6: Solve using substitution

Let \(x = a^2\), then the equation becomes a quadratic in \(x\):

\[
248x^2 - 62x - 4 = 0
\]

### Step 7: Solve the quadratic equation

Use the quadratic formula to solve for \(x\):

\[
x = \frac{-(-62) \pm \sqrt{(-62)^2 - 4(248)(-4)}}{2(248)}
\]

Simplify the terms:

\[
x = \frac{62 \pm \sqrt{3844 + 3968}}{496}
\]
\[
x = \frac{62 \pm \sqrt{7812}}{496}
\]

Approximate the square root:

\[
x = \frac{62 \pm 88.39}{496}
\]

Now, calculate the two solutions for \(x\):

1. \(x_1 = \frac{62 + 88.39}{496} = \frac{150.39}{496} \approx 0.303\)
2. \(x_2 = \frac{62 - 88.39}{496} = \frac{-26.39}{496} \approx -0.053\)

### Step 8: Solve for \(a\)

Since \(x = a^2\), we discard the negative value of \(x\) (since \(a^2\) cannot be negative), and we take the positive solution \(x_1 = 0.303\). Therefore:

\[
a^2 = 0.303
\]

Thus:

\[
a = \sqrt{0.303} \approx 0.55
\]

### Final Answer:
\[
a \approx 0.55
\]

Would you like more details on any step or have any other questions?

Here are 5 related questions:
1. How does the quadratic formula work in solving equations like this?
2. What are the conditions for discarding negative solutions when solving equations involving squares?
3. How can substitution help simplify higher-order equations?
4. What is the significance of the discriminant in the quadratic formula?
5. Can you solve similar equations using factorization instead of the quadratic formula?

**Tip:** Always check if you can simplify complex fractions early in the process to make the equation easier to solve.

Solve this: a^2 - \frac{169}{10404a^2} = \frac{13}{52}

Learn how to solve the quadratic equation a^2 - 169/(10404a^2) = 13/52 with a step-by-step solution, involving simplifying fractions, using substitution, and applying the quadratic formula. Suitable for students in Grades 10-12.

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