To solve the equation [tex]\( 25y^2 - 4x^2 - 250y + 24x + 489 = 0 \)[/tex] for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex], we follow several algebraic steps. Here is the detailed, step-by-step solution:
1. Start with the given equation:
[tex]\[
25y^2 - 4x^2 - 250y + 24x + 489 = 0
\][/tex]
2. Isolate the [tex]\( x \)[/tex]-related terms on one side:
[tex]\[
-4x^2 + 24x = -25y^2 + 250y - 489
\][/tex]
3. Factor out the common factor from the [tex]\( x \)[/tex]-related terms:
[tex]\[
-4(x^2 - 6x) = -25y^2 + 250y - 489
\][/tex]
4. Divide both sides of the equation by -4 to simplify:
[tex]\[
x^2 - 6x = \frac{25y^2 - 250y + 489}{4}
\][/tex]
5. Complete the square for the left side:
[tex]\[
x^2 - 6x + 9 = \frac{25y^2 - 250y + 489}{4} + 9
\][/tex]
This simplifies to:
[tex]\[
(x - 3)^2 = \frac{25y^2 - 250y + 525}{4}
\][/tex]
6. Rearrange the right side to prepare for taking the square root:
[tex]\[
(x - 3)^2 = \frac{25(y^2 - 10y + 21)}{4}
\][/tex]
7. Further simplify the right side:
[tex]\[
(x - 3)^2 = \frac{25(y - 7)(y - 3)}{4}
\][/tex]
8. Take the square root of both sides:
[tex]\[
x - 3 = \pm \sqrt{\frac{25(y - 7)(y - 3)}{4}}
\][/tex]
9. Simplify the square root:
[tex]\[
x - 3 = \pm \frac{5\sqrt{(y - 7)(y - 3)}}{2}
\][/tex]
10. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 3 \pm \frac{5\sqrt{(y - 7)(y - 3)}}{2}
\][/tex]
Thus, the solutions for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] are:
[tex]\[
x = 3 - \frac{5\sqrt{(y - 7)(y - 3)}}{2} \quad \text{or} \quad x = 3 + \frac{5\sqrt{(y - 7)(y - 3)}}{2}
\][/tex]
So, the two corresponding solutions are:
[tex]\[
(3 - \frac{5\sqrt{(y - 7)(y - 3)}}{2}, y) \quad \text{and} \quad (3 + \frac{5\sqrt{(y - 7)(y - 3)}}{2}, y)
\][/tex]
These are the pairs [tex]\((x, y)\)[/tex] that satisfy the given equation.
