Quadratic Formula Calculator

What this calculator does

So, you have a quadratic equation, ax² + bx + c = 0, and you want the values of x that make it true. This tool solves it from the three coefficients a, b, and c, and it copes with every kind of answer a quadratic can have: two different real solutions, one repeated solution, or a pair of complex ones.

You enter a, b, and c, and there is also a precision box that sets how many decimal places the answers are rounded to.

How to use it

1. Enter coefficient a, the number in front of x².
2. Enter b, the number in front of x, and c, the constant.
3. Optionally set the precision, the number of decimal places (it starts at 2).
4. Press Calculate.

The coefficient a cannot be zero. With no x² term, the equation is not a quadratic at all, it is a straight line, and there is nothing for this tool to solve.

The quadratic formula

Every quadratic is solved by one formula, and this is it:

x = ( -b ± √(b² - 4ac) ) / (2a)

Its great virtue is that it always works. Some quadratics factor neatly and can be solved by inspection, but many do not, and the formula does not care either way. The ± sign, read as plus or minus, is doing something important: it produces two answers at once, one using the plus and one using the minus, which is why a quadratic usually has two solutions.

The discriminant decides the three cases

The part tucked under the square root, b² - 4ac, is called the discriminant, and its sign settles the whole character of the answer before you finish the sum. The tool checks it first and branches on what it finds.

• If it is positive, the square root is a real number and the ± splits it into two, so you get two distinct real solutions.
• If it is zero, the square root is zero and the ± adds nothing, so the two solutions collapse into one repeated real solution.
• If it is negative, no real number squares to a negative, so the solutions are a pair of complex numbers.

If you only want that verdict, without solving, the discriminant calculator gives you the number on its own.

A worked example, step by step

Solve x² - 5x + 6 = 0, so a = 1, b = -5, c = 6.

• Discriminant: b² - 4ac = 25 - 24 = 1, which is positive, so expect two real solutions.
• Square root of 1 is 1.
• x = (5 ± 1) / 2, giving (5 + 1)/2 = 3 and (5 - 1)/2 = 2.

So the solutions are x = 3 and x = 2, the two values where the parabola crosses the x-axis.

What the complex solutions mean

When the discriminant is negative, the tool does not simply give up. No real number squares to a negative, so the answers involve i, the imaginary unit, defined as the square root of -1. The tool writes each solution as a real part plus or minus an imaginary part, in the form p ± qi. Take x² + x + 1 = 0: the discriminant is 1 - 4 = -3, and the solutions come out as -0.5 ± (roughly 0.87)i. There is nothing wrong here. It simply means the parabola never touches the x-axis, so its solutions live off the real number line.

Setting the precision

The precision box controls how many decimal places the solutions are rounded to, and it starts at 2. For a tidy answer like 3 and 2 that is plenty, but when a discriminant is not a perfect square, the roots run on as long decimals, and raising the precision lets you see more of them. It is a small control, but it means you can get a rough two-place answer or a sharper one from the same equation.

Questions people ask

Why can't a be zero?

Because without an x² term the equation is linear, not quadratic. The formula divides by 2a, and a of zero would leave you dividing by zero, so the tool asks for a nonzero a.

Why are there usually two solutions?

Because of the plus-or-minus in the formula. A parabola can cross the x-axis in two places, and those two crossings are the two solutions.

What does the discriminant tell me?

The nature of the solutions before you solve: positive means two real, zero means one repeated, negative means a complex pair. It is the sign of b² - 4ac.

What are complex solutions?

They appear when the discriminant is negative, and they involve i, the square root of -1. Geometrically, they mean the parabola never crosses the x-axis.

Should I just factor instead?

If the quadratic factors cleanly, that is quick. But many do not, and the formula solves them all, so it is the reliable fallback when factoring is not obvious.

References

On the formula and its long history. The quadratic formula solves every quadratic through the discriminant under its square root, and versions of the rule reach back well over a thousand years, with al-Khwarizmi setting it out around 820 alongside geometric completing-the-square proofs.

1. Eric W. Weisstein, "Quadratic Equation," from MathWorld, a Wolfram resource, on the quadratic, its formula, and its discriminant.
2. "Quadratic, cubic and quartic equations," MacTutor History of Mathematics, on al-Khwarizmi's rule for solving quadratics and its geometric roots.