To solve the equation $x(x - 2) = 5$, follow these steps:

Step 1: Expand the left side

Distribute $x$ across the terms inside the parentheses:

$x(x - 2) = x^2 - 2x$

So the equation becomes:

$x^2 - 2x = 5$

Step 2: Move all terms to one side

Now, subtract 5 from both sides to set the equation to 0:

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

Step 3: Solve the quadratic equation

You can solve the quadratic equation $x^2 - 2x - 5 = 0$ using the quadratic formula, which is:

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

For our equation, $a = 1$, $b = -2$, and $c = -5$.

Step 4: Apply the quadratic formula

Substitute $a = 1$, $b = -2$, and $c = -5$ into the formula:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-5)}}{2(1)}$

$x = \frac{2 \pm \sqrt{4 + 20}}{2}$

$x = \frac{2 \pm \sqrt{24}}{2}$

$x = \frac{2 \pm 2\sqrt{6}}{2}$

Step 5: Simplify the expression

Now simplify:

$x = 1 \pm \sqrt{6}$

So the two solutions are:

$x = 1 + \sqrt{6} \quad \text{or} \quad x = 1 - \sqrt{6}$

Final Answer:

The solutions are:

$x = 1 + \sqrt{6} \quad \text{or} \quad x = 1 - \sqrt{6}$

Would you like any further explanations or details about this process?

Related Questions:

1. How would you solve a quadratic equation with complex solutions?
2. Can you factor the quadratic equation $x^2 - 2x - 5 = 0$?
3. What is the discriminant of a quadratic equation, and how does it affect the solutions?
4. How do you check the solutions to a quadratic equation?
5. What happens if the discriminant is negative?

Tip: When solving quadratic equations, always check if factoring is possible before resorting to the quadratic formula. It can save time!