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Las soluciones para \( x \) son:
\[
x = 5 + \frac{\sqrt{80y - 480}}{2} \quad \text{y} \quad x = 5 - \frac{\sqrt{80y - 480}}{2}
\]
Solution
Para resolver la ecuación cuadrática \( x^{2}-10 x-20 y+145=0 \), primero identificamos que es una ecuación en términos de \( x \). La forma estándar de una ecuación cuadrática es \( ax^2 + bx + c = 0 \).
En este caso, podemos reescribir la ecuación como:
\[
x^{2} - 10x + (145 - 20y) = 0
\]
Aquí, tenemos:
- \( a = 1 \)
- \( b = -10 \)
- \( c = 145 - 20y \)
Ahora, podemos usar la fórmula cuadrática para encontrar las soluciones para \( x \):
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Sustituyendo los valores de \( a \), \( b \) y \( c \):
\[
x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot (145 - 20y)}}{2 \cdot 1}
\]
Simplificamos esto:
\[
x = \frac{10 \pm \sqrt{100 - 4(145 - 20y)}}{2}
\]
Ahora, calculamos el discriminante:
\[
100 - 4(145 - 20y) = 100 - 580 + 80y = 80y - 480
\]
Sustituyendo el discriminante en la fórmula:
\[
x = \frac{10 \pm \sqrt{80y - 480}}{2}
\]
Finalmente, simplificamos la expresión:
\[
x = 5 \pm \frac{\sqrt{80y - 480}}{2}
\]
Por lo tanto, las soluciones para \( x \) en términos de \( y \) son:
\[
x = 5 + \frac{\sqrt{80y - 480}}{2} \quad \text{y} \quad x = 5 - \frac{\sqrt{80y - 480}}{2}
\]
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Simplify this solution The Deep Dive
Did you know that the equation \( x^{2}-10 x-20 y+145=0 \) is a type of conic section? Specifically, it represents a parabola when rearranged into the standard form. Parabolas have fascinating properties, such as their axis of symmetry and vertex, which can often help in understanding their shape and position in a coordinate plane. If you're looking to solve this equation for \( y \), you can rearrange it to isolate \( y \): \( 20y = x^{2} - 10x + 145 \). Dynamic quadratic equations like these find their use in various fields, from physics and engineering to economics, where they can model projectile motion or the trajectory of objects.
