Quadratic Formula Calculator

ax2 + bx + c = 0
a = b = c =

Lesson on the Quadratic Formula

Lesson Contents

What is the Quadratic Formula?

The quadratic formula is a formula that is used to solve a quadratic equation of the standard form ax2 + bx + c = 0. The quadratic formula is given as:

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

Where x denotes the solution(s) to the quadratic equation, and a, b, and c are the coefficients of the quadratic equation. The part inside the square root (b2 – 4ac) is called the discriminant and it tells us how many roots the quadratic equation has.

How to Solve a Quadratic

To solve a quadratic equation, we must make sure it is in standard form. As shown earlier, standard form for a quadratic is ax2 + bx + c = 0. Once in standard form, we plug the equation’s coefficients into the quadratic formula.

If the discriminant is a positive number, continue solving until the two real solutions are obtained. Sometimes the discriminant will not be positive which results in us either taking the square root of zero or a negative number.

If the discriminant is zero, continue solving the quadratic formula as usual until the final, single solution is obtained. That final number is the solution, but we call it a repeated real solution. This means the graph of the quadratic touches the x-axis twice in the same spot.

If the discriminant is a negative number, taking the square root of it results in an imaginary number. Therefore, the quadratic equation has two imaginary solutions and no real solutions.

How the Calculator Works

The quadratic formula solver on this page is written in JavaScript (JS) and is powered by a JS native computer algebra system (CAS). The calculations all occur within your device’s internet browser JS engine, which allows for near-instantaneous answers and no waiting on the page to communicate with a server or refresh.

When the calculate button is pressed, your coefficients are fed to the CAS which then uses quadratic solving steps to symbolically calculate the solution(s). Being symbolic in nature, the CAS treats the quadratic like a human would with paper and pencil and maintains near-perfect accuracy.

When the CAS is done solving, the answer is converted to LaTeX (a math rendering language) and displayed in the answer area. If an error occurs or the inputs do not allow for a solution to be calculated, an error notice will instead be displayed.

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