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 right-hand side
Simplify the fraction on the right-hand side:

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

So, the equation becomes:

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

### Step 2: Simplify the left-hand side
Notice that \( \frac{169}{10404} \) simplifies to \( \frac{1}{61.5} \) or \( \frac{1}{62} \) approximately. But let’s first check the exact simplification:

\[
\frac{169}{10404} = \frac{169 \div 169}{10404 \div 169} = \frac{1}{61.5}
\]

Therefore, the equation becomes:

\[
a^2 - \frac{1}{61.5a^2} = \frac{1}{4}
\]

### Step 3: Multiply through by \( a^2 \)
Multiply both sides of the equation by \( a^2 \) to eliminate the denominator on the left:

\[
a^4 - \frac{1}{61.5} = \frac{a^2}{4}
\]

### Step 4: Clear the fractions
Now, multiply the entire equation by 61.5 to clear the denominator on the left:

\[
61.5a^4 - 1 = \frac{61.5a^2}{4}
\]

Multiply through:

\[
61.5a^4 - 1 = 15.375a^2
\]

### Step 5: Rearrange into standard quadratic form
Move all terms to one side:

\[
61.5a^4 - 15.375a^2 - 1 = 0
\]

This is now a quartic equation in terms of \( a^2 \). Let’s make a substitution to simplify things. Let \( x = a^2 \), so the equation becomes:

\[
61.5x^2 - 15.375x - 1 = 0
\]

### Step 6: Solve the quadratic equation
Solve this quadratic using the quadratic formula:

\[
x = \frac{-(-15.375) \pm \sqrt{(-15.375)^2 - 4(61.5)(-1)}}{2(61.5)}
\]

Simplify the terms inside the square root:

\[
x = \frac{15.375 \pm \sqrt{15.375^2 + 4(61.5)(1)}}{2(61.5)}
\]

\[
x = \frac{15.375 \pm \sqrt{236.390625 + 246}}{123}
\]

\[
x = \

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

This page provides a step-by-step solution to the quartic equation a^2 - (169/10404a^2) = 13/52, involving algebraic simplifications, solving a quadratic equation through substitution, and using the quadratic formula. Suitable for advanced high school students in Grades 10-12.

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