Enter the coefficients that form a quadratic equation to solve for x using the quadratic formula.

ax² + bx + c = 0

## Result:

x=-0.333333
x=-0.5

### Solution

6x²+5x+1=0
To solve, use the quadratic formula
x=-b ± b² - 4ac / 2a
Substitute your values for a, b, & c and solve
x=-5 ± 5² - 4·6·1 / 2·6
x=-5 ± 25 - 24 / 12
x=-5 ± 1 / 12
x=-5 ± 1 / 12
Then, reduce the problem
x=-5 / 12±1 / 12
x=-0.333333
x=-0.5
Learn how we calculated this below

## How to Use the Quadratic Formula

The quadratic formula can be used to calculate the solution for x in a quadratic equation. A quadratic equation is a second-degree polynomial equation.

The quadratic formula uses numerical coefficients from a quadratic equation to allow you to solve for the value of x. In the quadratic formula, a is the quadratic coefficient, b is the linear coefficient, and c is the constant.

You can solve a quadratic equation of the form ax² + bx + c = 0 using a quadratic equation solver like the one above or using the quadratic formula:

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

Thus, the value of x is equal to -b plus or minus the square root of b squared minus 4 times a times c over 2 times a, where a is not equal to 0.

When the discriminant b² – 4ac is equal to 0 then there will be a single solution for x, otherwise, there will be 2 possible solutions for x.

To use the quadratic formula, replace a, b, and c with the coefficients from the ax² + bx + c = 0 equation, then solve.

The value of a cannot be equal to zero as that would mean the formula is missing the , which would mean it’s not a quadratic.

Let’s solve the equation 3x² + 8x + 4 = 0 using the quadratic formula as an example.

Start by substituting the coefficients into the formula:

x = \frac{-8 \pm \sqrt{8^{2}-4 \cdot 3 \cdot 4}}{2 \cdot 3}

Begin solving the equation:

x = \frac{-8 \pm \sqrt{64-48}}{6}
x = \frac{-8 \pm \sqrt{16}}{6}
x = \frac{-8 \pm 4}{6}

Next, simplify the fraction:

x = \frac{-8}{6} \pm \frac{4}{6}
x = \frac{-4}{3} \pm \frac{2}{3}

Thus, there are two solutions for x:

x = \frac{-4}{3}+ \frac{2}{3}
x = \frac{-4}{3}- \frac{2}{3}

## How to Graph a Quadratic Equation

When graphed, the quadratic formula forms a parabola, or a curve in the shape of a U.

Note that smaller values of a form a wider parabolic curve, and larger values of a form a thinner parabola. A negative a value results in an upside-down parabola.

The resulting x values are the x-intercepts where the parabola crosses the x-axis.

### How to Find the Vertex

The general expression f(x) = ax² + bx + c is known as the standard form of a quadratic equation. But, you can also express a quadratic equation using vertex form.

A quadratic equation expressed in vertex form is:

f(x)=a(x-h)^{2}+k

Where:

h=\frac{b}{2a}
k = f(h)

The values of h and k in vertex form define the vertex point (h, k).

You might be interested in using our vertex form calculator to convert from standard form to vertex form.

### How to Find the Axis of Symmetry

The quadratic equation also defines an axis of symmetry, which is a vertical line at the center passing through the vertex of the parabola. The formula to calculate the axis of symmetry is:

x=\frac{b}{2a}

Thus, the x coordinate of the axis of symmetry is equal to negative b divided by 2 times a.