Quadratic Formula Calculator
The quadratic formula calculator gives both roots as decimals and in exact form (fractions or surds), plus the discriminant, the vertex and the factored form.
Roots
- Exact form
- x = 3 or x = 2
- Discriminant Positive, so two distinct real roots, and a perfect square, so both are rational.
- 1
- The formula applied
- x = (5 +/- sqrt(1)) / 2
- Vertex A minimum, since a is positive and the parabola opens upward.
- (2.5, -0.25)
- Axis of symmetry
- x = 2.5
- Factored form
- (x - 3)(x - 2) = 0
- Y-intercept
- (0, 6)
- Sum and product of the roots
- 5 and 6
- Completed square
- (x - 2.5)^2 - 0.25
How the Quadratic Formula Calculator works
Any equation of the form ax^2 + bx + c = 0, with a not zero, is solved by the quadratic formula: x = (-b plus or minus the square root of (b^2 - 4ac)) all over 2a. It always works, which is more than can be said for factoring.
The part under the root, b^2 - 4ac, is the discriminant, and it determines the shape of the answer before any arithmetic is done.
Positive: two distinct real roots, and the parabola crosses the x-axis twice. For x^2 - 5x + 6 the discriminant is 25 - 24 = 1, so the roots are (5 plus or minus 1) / 2, giving 3 and 2, and the factored form is (x - 3)(x - 2).
Zero: one repeated root, and the parabola touches the axis at a single point. x^2 - 4x + 4 has discriminant 0 and the root x = 2 twice, because it is (x - 2)^2.
Negative: no real roots, and the parabola misses the axis entirely. The two solutions are complex conjugates, same real part and opposite imaginary parts: x^2 + 1 = 0 gives x = i and x = -i.
The formula is not arbitrary; it is what completing the square produces. Divide through by a, move the constant across, add (b/2a)^2 to both sides to make the left side a perfect square, take the root of both sides. Doing that symbolically once gives the formula, and doing it to a specific equation gives the vertex form: a(x - h)^2 + k, where h = -b/2a and k is the value of the quadratic there.
Which is where the vertex comes from. The axis of symmetry is x = -b/2a, the same -b/2a in the formula, because the two roots sit symmetrically either side of it at a distance of sqrt(discriminant)/2a. The vertex is the point on the curve at that x, which makes quadratics the simplest useful optimization problem: maximum revenue, maximum projectile height, minimum cost.
Vieta's formulas give a quick check that costs nothing. The sum of the roots is -b/a and the product is c/a. For x^2 - 5x + 6, the sum should be 5 and the product 6: 3 + 2 = 5 and 3 x 2 = 6. They hold for complex roots too, which makes them the fastest way to check an answer.
If a = 0 the equation is not quadratic at all: it is bx + c = 0, with the single root -c/b, and the quadratic formula would divide by zero. If b is also zero there is nothing left to solve.
Quadratic Formula Calculator FAQ
What is the quadratic formula?
x = (-b plus or minus sqrt(b^2 - 4ac)) / 2a, for any equation ax^2 + bx + c = 0 with a not zero. It comes from completing the square on the general form, and it works on every quadratic, including those that cannot be factored.
What does the discriminant tell you?
b^2 - 4ac decides the nature of the roots. Positive means two distinct real roots; zero means one repeated root with the vertex on the x-axis; negative means no real roots and a complex conjugate pair. If it is a positive perfect square, the roots are rational and the quadratic factors over the integers.
How do I find the vertex of a parabola?
The x-coordinate is -b/2a, the same expression that appears in the quadratic formula, because the roots are symmetric about it. Substitute that back into the equation for the y-coordinate. It is a minimum when a is positive and a maximum when a is negative.
Why can't I always factor a quadratic?
Factoring over the integers only works when the discriminant is a perfect square. x^2 - 2x - 1 has a discriminant of 8, so its roots are 1 plus or minus sqrt(2). Those are irrational, and no integer factorization exists. The quadratic formula works regardless, which is why it is the general method.