## In math, how to solve $x^2 - 2x -4 <= 0$

To solve the inequality $x^2 - 2x - 4 \leq 0$, you need to determine the values of $x$ for which the quadratic expression is less than or equal to zero.

First, solve the quadratic equation $x^2 - 2x - 4 = 0$ to find its roots. The quadratic formula is given by:

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

In this case, $a = 1$, $b = -2$, and $c = -4$. Substituting these values into the quadratic formula:

$$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1}$$
$$x = \frac{2 \pm \sqrt{4 + 16}}{2}$$
$$x = \frac{2 \pm \sqrt{20}}{2}$$
$$x = \frac{2 \pm 2\sqrt{5}}{2}$$
$$x = 1 \pm \sqrt{5}$$

So, the roots are $x = 1 + \sqrt{5}$ and $x = 1 - \sqrt{5}$.

Next, determine the intervals defined by these roots. The roots split the number line into three intervals:

1. $(-\infty, 1 - \sqrt{5})$
2. $[1 - \sqrt{5}, 1 + \sqrt{5}]$
3. $(1 + \sqrt{5}, \infty)$

Determine the sign of the quadratic expression in each interval by testing a point within each interval.

1. For $(-\infty, 1 - \sqrt{5})$, select $x = -10$:
  \[(-10)^2 - 2(-10) - 4 = 100 + 20 - 4 = 116 \]

2. For $(1 - \sqrt{5}, 1 + \sqrt{5})$, select $x = 0$:
  \[(0)^2 - 2(0) - 4 = -4 \]

3. For $(1 + \sqrt{5}, \infty)$, select $x = 10$:
  \[10^2 - 2(10) - 4 = 100 - 20 - 4 = 76 \]

Considering these results, the quadratic expression is negative or zero in the interval where $x$ lies between the roots $1 - \sqrt{5}$ and $1 + \sqrt{5}$ inclusive. Thus, the solution is:

$$ x \in [1 - \sqrt{5}, 1 + \sqrt{5}] $$

> Model used: gpt-4o-mini