Quadratic Formula Calculator

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
Next, solve the radical
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


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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.

Graphic showing the quadratic formula used to solve quadratic equations

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.

Graph of a quadratic equation showing the x-intercepts, vertex, and axis of symmetry

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.