Questions: 5/(x^2-5x) - 1/(x^2-25) = 0

Solution Steps

To solve the equation \(\frac{5}{x^{2}-5x}-\frac{1}{x^{2}-25}=0\), we need to find a common denominator and then solve for \(x\). The common denominator for the fractions is \((x^2 - 5x)(x^2 - 25)\). We will then set the numerators equal to each other and solve the resulting equation.

We start with the equation

\[ \frac{5}{x^{2}-5x}-\frac{1}{x^{2}-25}=0. \]

The common denominator for the fractions is \((x^2 - 5x)(x^2 - 25)\). We rewrite the equation as:

\[ \frac{5(x^2 - 25) - 1(x^2 - 5x)}{(x^2 - 5x)(x^2 - 25)} = 0. \]

Setting the numerator equal to zero gives us:

\[ 5(x^2 - 25) - (x^2 - 5x) = 0. \]

Expanding this, we have:

\[ 5x^2 - 125 - x^2 + 5x = 0, \]

which simplifies to:

\[ 4x^2 + 5x - 125 = 0. \]

Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 4\), \(b = 5\), and \(c = -125\), we find the solutions. The discriminant is calculated as:

\[ b^2 - 4ac = 5^2 - 4 \cdot 4 \cdot (-125) = 25 + 2000 = 2025. \]

Thus, the solutions are:

\[ x = \frac{-5 \pm \sqrt{2025}}{2 \cdot 4}. \]

Calculating \(\sqrt{2025} = 45\), we have:

\[ x = \frac{-5 \pm 45}{8}. \]

This gives us two potential solutions:

1. \(x = \frac{40}{8} = 5\)
2. \(x = \frac{-50}{8} = -\frac{25}{4}\)

Final Answer