Easy Guide to Using the Quadratic Formula

The Quadratic Formula is a useful way to find the answers (or roots) of any quadratic equation. A quadratic equation usually looks like this:

$ax^2 + bx + c = 0$

Here, $a$, $b$, and $c$ are numbers, and $a$ can’t be zero. The formula to find $x$ is:

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

Let’s break down the steps to use the Quadratic Formula:

1. Find the Numbers (Coefficients):

   • In the equation $ax^2 + bx + c = 0$, look for the values of $a$, $b$, and $c$.
   • For example, in $2x^2 + 4x - 6 = 0$, we have $a = 2$, $b = 4$, and $c = -6$.

2. Calculate the Discriminant:

   • The discriminant helps us understand the type of roots we will get. We use the formula: $D = b^2 - 4ac$
   • For our example: $D = 4^2 - 4 \cdot 2 \cdot (-6) = 16 + 48 = 64$

3. See What the Discriminant Tells Us:

   • If $D > 0$: We have two different real roots.
   • If $D = 0$: There is one real root (also called a double root).
   • If $D < 0$: We get two complex (non-real) roots.
   • In our case, since $D = 64 > 0$, we’ll have two different real roots.

4. Put the Values into the Formula:

   • Now plug the numbers ($a$, $b$, and the square root of $D$) into the formula.
   • For our example: $x = \frac{-4 \pm \sqrt{64}}{2 \cdot 2}$

5. Make the Expression Simpler:

   • First, find the square root, then simplify. For our example: $x = \frac{-4 \pm 8}{4}$

6. Find the Two Possible Values for x:

   • Now, calculate both options:
     • For $x_1$: $x_1 = \frac{-4 + 8}{4} = \frac{4}{4} = 1$
     • For $x_2$: $x_2 = \frac{-4 - 8}{4} = \frac{-12}{4} = -3$

7. Wrap Up the Solutions:

   • The solutions to the equation $2x^2 + 4x - 6 = 0$ are $x = 1$ and $x = -3$.

By following these simple steps, you can easily use the Quadratic Formula to solve any quadratic equation. This method not only helps you find the answers but also strengthens your understanding of quadratic functions!