Quadratic Formula Calculator
Enter any quadratic equation in standard, vertex, or factored form and instantly get both roots (real or complex), the discriminant, the vertex coordinates, the axis of symmetry, and the equation rewritten in all three standard forms. The show-your-work panel walks through every calculation so you can follow along or check your homework.
Formula
Worked example
For x^2 - 3x + 2 = 0 (a=1, b=-3, c=2): discriminant = 9 - 8 = 1, sqrt(1) = 1, x1 = (3+1)/2 = 2, x2 = (3-1)/2 = 1. Vertex at (1.5, -0.25). Factored form: (x-2)(x-1).
How the calculator works
The calculator accepts your quadratic equation in whichever form you already have it: standard form (ax^2 + bx + c = 0), vertex form (a(x-h)^2 + k = 0), or factored form (a(x-r1)(x-r2) = 0). Vertex and factored inputs are algebraically expanded into standard coefficients first, then the quadratic formula is applied. The discriminant b^2 - 4ac is computed before any roots, because its sign immediately tells you whether to expect two real roots, a single repeated root, or a complex-conjugate pair.
Vertex, axis of symmetry, and extra forms
Beyond the roots, the calculator also gives you the vertex of the parabola at h = -b/(2a) and k = c - b^2/(4a). The axis of symmetry is the vertical line x = h that divides the parabola into two mirror halves. When the discriminant is non-negative, the equation is also displayed in vertex form a(x-h)^2 + k and factored form a(x-x1)(x-x2), which are often needed for graphing or further manipulation. Toggle "Show extra forms" to see those outputs.
Vieta's formulas: a fast sanity check
Vieta's formulas relate the coefficients directly to the roots without any square roots: the sum of the two roots is always -b/a, and their product is c/a. These hold even for complex roots. They are the fastest way to check your answer: if your two roots do not add to -b/a or multiply to c/a, something went wrong. They are also useful when you only need the sum or product of the roots and do not need the roots themselves.
When to use each input form
Use standard form when you have coefficients written out as ax^2 + bx + c. Use vertex form when the equation is already written as a(x-h)^2 + k, which is common in graphing problems where the vertex is given. Use factored form when you know both roots and the leading coefficient, perhaps from a graph or a partially solved problem, and you need to recover the standard coefficients or verify the discriminant. All three paths produce identical results.
Discriminant and root types
| Discriminant (delta) | Root type | Parabola |
|---|---|---|
| delta > 0 | Two distinct real roots | Crosses x-axis at two points |
| delta = 0 | One repeated real root | Touches x-axis at one point (tangent) |
| delta < 0 | Two complex-conjugate roots | Does not cross the x-axis |
The sign of b^2 - 4ac fully determines how many real solutions the equation has.
Frequently asked questions
What does a negative discriminant mean?
A negative discriminant means the quadratic has no real roots. The two solutions are complex conjugates of the form p + qi and p - qi, where p = -b/(2a) is the real part and q involves the square root of the absolute discriminant value. Geometrically, the parabola does not cross or touch the x-axis. Complex roots always appear as conjugate pairs when the original coefficients are real numbers.
Can a quadratic have only one root?
Yes. When the discriminant equals exactly zero, the quadratic formula produces one repeated root at x = -b/(2a). This is called a double root or a root of multiplicity two. Geometrically, the parabola is tangent to the x-axis at exactly one point, which is also the vertex.
Why must a be nonzero?
If a = 0, the x^2 term disappears and the equation reduces to bx + c = 0, which is linear with at most one solution. The quadratic formula divides by 2a, so a = 0 causes division by zero. The calculator returns a blank result rather than a meaningless number when a is zero.
What is the vertex of a parabola and why does it matter?
The vertex is the point where the parabola reaches its minimum value (when a > 0, opens upward) or its maximum value (when a < 0, opens downward). Its x-coordinate is -b/(2a), which is also the axis of symmetry. For optimization problems, the vertex gives the minimum or maximum of the quadratic function directly, without needing to find the roots.
What are Vieta's formulas?
Vieta's formulas state that for ax^2 + bx + c = 0 with roots x1 and x2, the sum x1 + x2 equals -b/a and the product x1 * x2 equals c/a. These identities come from expanding a(x - x1)(x - x2) and matching coefficients. They let you check computed roots instantly: if they do not sum to -b/a, at least one root is wrong. They also let you construct a quadratic from its roots: set a = 1, b = -(sum), c = product.
What is the difference between factored form and vertex form?
Factored form a(x - x1)(x - x2) makes the roots immediately visible as x1 and x2, so it is ideal when you need to find where the parabola crosses the x-axis. Vertex form a(x - h)^2 + k makes the vertex (h, k) immediately visible and is easier for graphing or completing-the-square problems. Both can be converted to standard form ax^2 + bx + c by expanding, which is what this calculator does when you select a non-standard input mode.
How do I use this calculator for a projectile motion problem?
Projectile height is typically modeled as h(t) = -4.9t^2 + v0*t + h0 (SI) or h(t) = -16t^2 + v0*t + h0 (US customary, in feet), where v0 is initial speed and h0 is initial height. To find when the projectile lands, set h = 0 and read off a = -4.9 (or -16), b = v0, c = h0, then enter those into the standard form mode. The positive real root gives the landing time.