Let's solve
$$x^2-3x-4=0$$
with the quadratic formula:
$$x=\frac{-\textcolor{#5cb85c}{b} \pm \sqrt{\textcolor{#5cb85c}{b}^2-4\textcolor{#2d6da3}{a}\textcolor{#d9534f}{c}}}{2\textcolor{#2d6da3}{a}}$$
There are three steps.
Step 1: Find a, b, c
Let's first find the numbers a, b, and c.
We can rewrite our equation as:
$$\textcolor{#2d6da3}{1}x^2+\textcolor{#5cb85c}{-3}x+\textcolor{#d9534f}{-4}=0$$
And then we see that:
$$\textcolor{#2d6da3}{a=1}$$
$$\textcolor{#5cb85c}{b=-3}$$
$$\textcolor{#d9534f}{c=-4}$$
Step 2: Plug in a, b, c
Let's now plug a, b, and c into the quadratic formula. We'll replace the a's with 1, the b’s with -3, and the c's with -4.
$$x=\frac{-\textcolor{#5cb85c}{b} \pm \sqrt{\textcolor{#5cb85c}{b}^2-4\textcolor{#2d6da3}{a}\textcolor{#d9534f}{c}}}{2\textcolor{#2d6da3}{a}}$$
$$x=\frac{-\textcolor{#5cb85c}{(-3)} \pm \sqrt{\textcolor{#5cb85c}{(-3)}^2-4\textcolor{#2d6da3}{(1)}\textcolor{#d9534f}{(-4)}}}{2\textcolor{#2d6da3}{(1)}}$$
Step 3: Simplify
And now let's simplify. We'll simplify this formula piece by piece. The negative of negative three, is three.
$$x=\frac{3 \pm \sqrt{\textcolor{#5cb85c}{(-3)}^2-4\textcolor{#2d6da3}{(1)}\textcolor{#d9534f}{(-4)}}}{2\textcolor{#2d6da3}{(1)}}$$
On the bottom we have two times one, which simplifies to two.
$$x=\frac{3 \pm \sqrt{\textcolor{#5cb85c}{(-3)}^2-4\textcolor{#2d6da3}{(1)}\textcolor{#d9534f}{(-4)}}}{2}$$
Now inside the radical, let's start with negative three squared, which is 9.
$$x=\frac{3 \pm \sqrt{9-4\textcolor{#2d6da3}{(1)}\textcolor{#d9534f}{(-4)}}}{2}$$
Then four times one, is four:
$$x=\frac{3 \pm \sqrt{9-(4)\textcolor{#d9534f}{(-4)}}}{2}$$
Four times negative four is negative 16:
$$x=\frac{3 \pm \sqrt{9-(-16)}}{2}$$
9 minus negative 16 is equal to 25:
$$x=\frac{3 \pm \sqrt{25}}{2}$$
The square root of 25 is 5:
$$x=\frac{3 \pm 5}{2}$$
And now let's break up the plus and minus into two answers:
$$x=\frac{3 + 5}{2} \text{\enspace\enspace or \enspace\enspace} x=\frac{3 - 5}{2}$$
Let's simplify each answer. So three plus five is equal to 8:
$$x=\frac{8}{2} \text{\enspace\enspace or \enspace\enspace} x=\frac{3 - 5}{2}$$
8 divided by 2 is equal to 4:
$$x=4 \text{\enspace\enspace or \enspace\enspace} x=\frac{3 - 5}{2}$$
So our first answer is x is equal to 4.
For our second solution, we see that three minus five is equal to negative two:
$$x=4 \text{\enspace\enspace or \enspace\enspace} x=\frac{-2}{2}$$
And then negative two over two is equal to negative one.
Answer
So the final answer is X is equal to 4, or X is equal to negative 1:
$$x=4 \text{\enspace\enspace or \enspace\enspace} x=-1$$
More Examples
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