Math Snap

STEP 1

What is this asking?
We need to find the values of $k$ that make the equation $k^2 - k - 4 = 0$ true, using the quadratic formula!
Watch out!
Remember the quadratic formula has a plus-or-minus sign, meaning we'll likely have two answers!
Also, don't forget to simplify your final answer as much as possible.

STEP 2

1. Identify the coefficients.
2. Apply the quadratic formula.
3. Simplify the result.

STEP 3

Let's define our quadratic equation as $ak^2 + bk + c = 0$.
Looking at our given equation, $k^2 - k - 4 = 0$, we can match it up to see what $a$, $b$, and $c$ are.

STEP 4

The coefficient of $k^2$ is $\bf{1}$, so $a = \bf{1}$.
The coefficient of $k$ is $\bf{-1}$, so $b = \bf{-1}$.
The constant term is $\bf{-4}$, so $c = \bf{-4}$.

STEP 5

Remember, the quadratic formula is given by:
$$k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Let's plug in our values for $a$, $b$, and $c$.

STEP 6

Substituting $a = \bf{1}$, $b = \bf{-1}$, and $c = \bf{-4}$, we get:
$$k = \frac{-(\bf{-1}) \pm \sqrt{(\bf{-1})^2 - 4 \cdot \bf{1} \cdot (\bf{-4})}}{2 \cdot \bf{1}}$$

STEP 7

Let's simplify inside the square root first:
$$k = \frac{1 \pm \sqrt{1 + 16}}{2}$$ $$k = \frac{1 \pm \sqrt{17}}{2}$$

STEP 8

Since $\sqrt{17}$ cannot be simplified further, our two solutions are:
$$k = \frac{1 + \sqrt{17}}{2} \quad \text{or} \quad k = \frac{1 - \sqrt{17}}{2}$$

STEP 9

As decimals rounded to the nearest hundredth, these are approximately:
$$k \approx \frac{1 + 4.12}{2} \approx \frac{5.12}{2} \approx 2.56$$ $$k \approx \frac{1 - 4.12}{2} \approx \frac{-3.12}{2} \approx -1.56$$

$k = \frac{1 + \sqrt{17}}{2}$ or $k = \frac{1 - \sqrt{17}}{2}$.
As decimals, $k \approx 2.56$ or $k \approx -1.56$.