Math Snap

STEP 1

What is this asking?
We need to find the values of $x$ that make this specific quadratic equation equal to zero.
Watch out!
Don't forget there are two solutions to most quadratic equations, thanks to the plus-or-minus in the quadratic formula!

STEP 2

1. Identify Coefficients
2. Apply the Quadratic Formula
3. Simplify the Expression
4. Calculate the Solutions

STEP 3

Alright, let's kick things off by looking at our equation: $x^2 - 5x - 24 = 0$.
Remember, the standard form of a quadratic equation is $ax^2 + bx + c = 0$.

STEP 4

So, matching things up, we can see that our coefficients are $a = 1$, $b = -5$, and $c = -24$. Crucially, don't forget that negative sign in front of the 5 and 24!

STEP 5

Now, let's bring in the star of the show: the quadratic formula itself! $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

STEP 6

Let's plug in those coefficients we just found: $a = 1$, $b = -5$, and $c = -24$.
$$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1}$$

STEP 7

Time to clean things up!
First, that double negative in front of the 5 becomes a positive 5.
$$x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1}$$

STEP 8

Next, inside the square root, $(-5)^2$ is $(-5) \cdot (-5)$, which gives us positive 25.
Then, $4 \cdot 1 \cdot (-24)$ gives us -96.
$$x = \frac{5 \pm \sqrt{25 - (-96)}}{2}$$

STEP 9

Subtracting a negative is the same as adding, so $25 - (-96)$ becomes $25 + 96 = 121$.
$$x = \frac{5 \pm \sqrt{121}}{2}$$

STEP 10

The square root of 121 is 11, because $11 \cdot 11 = 121$.
$$x = \frac{5 \pm 11}{2}$$

STEP 11

Almost there!
Now we split the plus-or-minus into two separate calculations.
$$x = \frac{5 + 11}{2} \quad \text{or} \quad x = \frac{5 - 11}{2}$$

STEP 12

For the first one, $\frac{5 + 11}{2} = \frac{16}{2} = 8$.
$$x = 8$$

STEP 13

For the second one, $\frac{5 - 11}{2} = \frac{-6}{2} = -3$.
$$x = -3$$

So, our two solutions are $x = 8$ and $x = -3$!