Quadratic Formula Example


Video Transcript

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.


So the final answer is X is equal to 4, or X is equal to negative 1: $$x=4 \text{\enspace\enspace or \enspace\enspace} x=-1$$

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