Quadratic Formula Calculator
Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 with the quadratic formula. Enter a, b and c to get both roots in exact and decimal form, real or complex, with the discriminant, the vertex, the factored form, every step of the working and a graph of the parabola.
Graph of the parabola
What the discriminant says
x = (-b ± √(b² – 4ac)) / 2a. The discriminant b² – 4ac decides the roots: positive gives two real roots, zero gives one repeated root and negative gives two complex roots. Exact forms are shown when a, b and c are whole numbers, with square roots simplified and fractions reduced; decimal roots use a form of the formula that keeps full accuracy even when one root is tiny. If a is 0 the equation is linear and has at most one solution. The CSV, image and printed report contain the values you entered, nothing preloaded.
How to use this tool
Write the equation as ax² + bx + c = 0 and enter the three numbers, including any minus signs. The roots appear at the top as they would be written by hand, with fractions and square roots simplified, and in decimals below. If a, b and c are whole numbers you also get the factored form when there is one.
The steps box shows the formula worked through with your numbers. The graph marks the roots where the curve crosses the x axis, the vertex and the y intercept, and the table lists values either side of the vertex so you can plot it yourself.
How the quadratic formula works
Any quadratic ax² + bx + c = 0 can be rearranged by completing the square into (x + b/2a)² = (b² – 4ac)/4a². Taking the square root of both sides and moving b/2a across gives x = (-b ± √(b² – 4ac)) / 2a, the formula every method comes back to.
The discriminant, b² – 4ac, is the part under the square root. When it is positive there are two real roots, when it is zero the two roots meet at the vertex, and when it is negative the square root is imaginary and the roots are a complex pair. The roots always add up to -b/a and multiply to c/a, a quick check on any answer.
A worked example you can check on paper
Solve 2x² – 5x – 3 = 0. Here a = 2, b = -5 and c = -3. The discriminant is (-5)² – 4 x 2 x (-3) = 25 + 24 = 49, and √49 = 7.
Then x = (5 ± 7) / 4. With the plus sign, x = 12/4 = 3; with the minus sign, x = -2/4 = -1/2. As a check, 3 + (-1/2) = 5/2, which is -b/a, and 3 x (-1/2) = -3/2, which is c/a. The factored form is (2x + 1)(x – 3) = 0.
Mistakes that catch people out
Dropping a minus sign. In x² – 5x + 6, b is -5, so -b is +5. Squaring a negative b always gives a positive number.
Dividing only part of the top by 2a. The whole of -b ± √D sits over 2a, not just the square root.
Not setting the equation to zero. 3x² = 2x + 5 has to become 3x² – 2x – 5 = 0 before a, b and c can be read off.
Stopping at a negative discriminant. No real roots is not the same as no roots; the complex pair still solves the equation.
Popular calculations
Each opens the calculator with that equation solved, and every number can be changed once it is open.
Equations that factor
Perfect squares and differences of squares
Irrational roots
Complex roots
Decimals and projectile problems
How this was checked
The engine was tested in Node on more than twenty equations covering two, one and no real roots, fractions, surds, zero constants and decimal coefficients. Every real root was put back into its equation and gave zero to within rounding, and the exact forms were compared with hand working, such as (1 ± √5)/2 for x² – x – 1 = 0 and -3/4 ± (√23/4)i for 2x² + 3x + 4 = 0.
Every address in the lists above was then opened and checked for a distinct title, the right roots, charts that render, and clean CSV, image and print output containing only that equation, at desktop, tablet and phone widths.
Common questions
What is the quadratic formula?
For any equation ax² + bx + c = 0 with a not zero, x = (-b ± √(b² – 4ac)) / 2a. The plus sign gives one root and the minus sign the other. It comes from completing the square on the general equation.
How do you use the quadratic formula?
Move every term to one side so the other side is 0, read off a, b and c with their signs, work out b² – 4ac, then put the numbers into the formula. For 2x² – 5x – 3 = 0, the discriminant is 25 + 24 = 49, so x = (5 ± 7)/4, giving 3 and -1/2.
What does the discriminant tell you?
b² – 4ac tells you what kind of roots to expect before you finish. Positive means two different real roots, zero means one repeated root where the parabola just touches the x axis, and negative means two complex roots and no crossing at all.
What if the discriminant is negative?
The square root of a negative number is imaginary, so the roots are a complex pair p ± qi. For x² + 2x + 5 = 0 the discriminant is -16, √-16 is 4i, and x = (-2 ± 4i)/2 = -1 ± 2i.
Can every quadratic be factored?
With whole number coefficients, it factors neatly over whole numbers and fractions only when the discriminant is a perfect square. x² – x – 1 = 0 has discriminant 5, so its roots are (1 ± √5)/2 and the formula is the practical way to solve it.
How do I find the vertex of a parabola?
The vertex is at x = -b/2a, halfway between the roots, and its height is c – b²/4a. For x² – 5x + 6 that is x = 2.5 and y = -0.25, the lowest point because a is positive.
Related Calculators and Reference
- Square Root Calculator: square roots with the radical simplified.
- Fraction Calculator: add, subtract and simplify fractions.
- Slope Calculator: slope and equation of a straight line.
- Scientific Calculator: any expression with the working shown.
- System of Equations Calculator: solve two or three linear equations at once.
- Complex Number Calculator: add, multiply and convert the complex roots that come from a negative discriminant.
- Wolfram MathWorld: Quadratic Formula: the formula, its derivation and history.