Calculus > How to Use the Quadratic Formula in Calculus

The quadratic formula is a way to find the solution for any polynomial in the form ax^{2} + bx + c = 0. You’ll need to use the quadratic formula to find the solutions for polynomials in many places; for example, you can use solutions for polynomials to find total distance for velocity equations. It works when factoring doesn’t, or when factoring is too complicated. In fact, the formula will *always* give you a solution, while factoring sometimes works and sometimes doesn’t.

The quadratic formula is stated as:

*For any function of the form ax ^{2} + bx + c = 0, the value of x is given by:*

“a”, “b” and c are just numbers, or numerical coefficients. The formula is derived from completing the square.

## Quadratic Formula Example

**Sample problem:** Solve x^{2} + 3x + 4 using the quadratic formula.

Step 1: **Identify a, b and c **in your function.

We know that the quadratic formula is: ax^{2} + bx + c = 0. So, in this example, a=1, b=3 and c=4. Note that there isn’t a number before x^{2}, so we put a “1.” That’s because 1 * x^{2} = x^{2}.

Step 2:** Insert a, b, and c** into the quadratic formula:

**Use algebra**to solve the formula:

This further simplifies to -8/2, 2/2, or -4, 1, which is the solution for this particular quadratic equation.

**Tip:** In order for the formula to work, your function must be in the form ax^{2} + bx + c = 0. You may need to use a little algebra to get your function in the right form. Note that the “2a” in the denominator is underneath *everything *, not just the square root. Most mistakes are made by accidentally dropping the square root, putting “a” in the denominator instead of “2a” or dropping the “plus/minus.” Also, when you figure out “b^{2},” the answer is always going to be positive, even if b has a negative before it. Don’t forget to include signs when working through the equation.

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