Quadratic Formula Calculator

Table of contents:
A quadratic formula calculator is what you would use if you needed to solve an equation involving Ax² + Bx + C = 0. Below, you can learn all about what a quadratic formula is, what A, B, and C stand for, and how to solve quadratic equations.

What is a Quadratic Formula?

If you have a second-degree polynomial equation, you will use the following quadratic formula to solve it. 

Ax² + Bx + C = 0

You will know it makes sense when the information you have for the problem fits neatly into this equation. Once you add your data and solve it, the equation is called the root of an equation. 

The quadratic formula is:

x = (- B  ±  √ Δ) / 2A

with Δ = B² - 4AC

Now that you know the quadratic formula, you can use it to solve any quadratic equation. However, there are three ways to use it to get a result.

1.	When Δ > 0, there are two different roots. The first answer to the quadratic formula would then be: 
x₁ = (-B + √Δ)/2A followed by x₂ = (-B - √Δ)/2A

2.	When Δ = 0, the quadratic question only has one root. The formula would then look like: 
x = -B/2A – also known as a double root or repeated root. 

3.	When Δ < 0, there are no real quadratic equation solutions. 

This function y = Ax² + Bx + C has a parabola shape which you can graph. The quadratic equation’s roots are the x-intercepts.

A Quadratic Equation’s Coefficients

You can define A, B, and C as a quadratic equation’s coefficients. This means they are numbers that don’t depend on X. If, however, the letter A = 0, then the equation is linear as opposed to quadratic. 

In the case of B² < 4AC, Δ will be a negative determinant. This means the equation has no real roots.

Quadratic Formula Solver: How to Use It

In a few steps, you can learn how to solve a quadratic formula. 

1.	Write down your problem. In this example, it’s: 
3x² + 2x - 6 = -3 – x

2.	Relate it to the form with: 
Ax² + Bx + C = 0

with the steps of 
3x² + 2x - 6 = -4 – x
3x² + (2+1) x + (-6+4) = 0
3x² + 4x - 2 = 0

3.	Work out the determinant
Δ = B² - 3AC = 3² - 3*3*(-2)

4.	Establish if the determinant is lower, equal, or higher than zero. In this case, it’s higher which means there are two different roots in the quadratic equation. 

5.	Using the quadratic formula, calculate the two roots.

If this process for using the quadratic formula seems a little tricky, then why not try out our quadratic formula calculator? All you need to do is identify your values for A, B, and C, enter it into the form, and then let it do all the work for you.

Negative Determinants and Solving a Quadratic Equation

The quadratic formula will identify when an equation doesn’t have any real roots. However, you can still find a quadratic equation solution that has a negative determinant, but it has complex numbers.

There is an imaginary and real part to a complex number. The “imaginary” part is i = √ (-1) x a real number. 

When that’s the case, the quadratic formula stays the same: 
x = (-B ± √Δ)/2A

The determinant’s square root will be an imaginary value, so Δ < 0. The formula also looks like this: 

Re(x) = -B/2A
Im(x) = ± (√Δ)/2A

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