X^2 + X - 30
___________
X - 5
= 1

Q: X^2 + X - 30 ___________ X - 5 = 1

To solve the equation \[ \frac{x^2 + x - 30}{x - 5} = 1 \] follow these steps: 1. Multiply both sides by $x - 5$ to eliminate the fraction: \[ x^2 + x - 30 = x - 5 \] 2. Subtract $x$ and add 5 to both sides to simplify: \[ x^2 - 30 + 5 = 0 \] \[ x^2 - x - 25 = 0 \] 3. Factor the quadratic equation: The equation does not factor easily, so use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where $a = 1$, $b = -1$, and $c = -25$. 4. Calculate the discriminant: \[ b^2 - 4ac = (-1)^2 - 4 \cdot 1 \cdot (-25) = 1 + 100 = 101 \] 5. Solve for $x$: \[ x = \frac{-(-1) \pm \sqrt{101}}{2 \cdot 1} \] \[ x = \frac{1 \pm \sqrt{101}}{2} \] So the solutions are: \[ x = \frac{1 + \sqrt{101}}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{101}}{2} \]

Question

Accepted Answer

To solve the equation

$$
\frac{x^2 + x - 30}{x - 5} = 1
$$

follow these steps:

1. Multiply both sides by $x - 5$ to eliminate the fraction:

$$
   x^2 + x - 30 = x - 5
   $$

2. Subtract $x$ and add 5 to both sides to simplify:

$$
   x^2 - 30 + 5 = 0
   $$

$$
   x^2 - x - 25 = 0
   $$

3. Factor the quadratic equation:

The equation does not factor easily, so use the quadratic formula:

$$
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   $$

where $a = 1$, $b = -1$, and $c = -25$.

4. Calculate the discriminant:

$$
   b^2 - 4ac = (-1)^2 - 4 \cdot 1 \cdot (-25) = 1 + 100 = 101
   $$

5. Solve for $x$:

$$
   x = \frac{-(-1) \pm \sqrt{101}}{2 \cdot 1}
   $$

$$
   x = \frac{1 \pm \sqrt{101}}{2}
   $$

So the solutions are:

$$
x = \frac{1 + \sqrt{101}}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{101}}{2}
$$