Quadratic Formula Calculator

What does a quadratic formula calculator do?

A quadratic formula calculator solves any equation written as ax² + bx + c = 0, returning its roots, the discriminant, and the vertex of the matching parabola. You enter the three coefficients — a, b and c — and the tool applies the quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a to find every value of x that makes the equation true.

A quadratic is a second-degree polynomial: the highest power of x is 2. Its graph is a parabola, a U-shaped curve, and the roots are exactly the points where that curve crosses the x-axis. Depending on the numbers, a quadratic can have two real roots, one repeated root, or two complex roots — and this calculator detects which case you are in automatically.

How does the quadratic formula work?

Everything hinges on one expression inside the square root, called the discriminant:

D = b² − 4ac

The discriminant decides the number and type of roots before you finish the calculation:

• D greater than 0 → two distinct real roots, given by (−b ± √D) ÷ 2a.
• D equal to 0 → one repeated real root, x = −b ÷ 2a.
• D less than 0 → two complex roots, −b/2a ± (√−D ÷ 2a)i, where i is the imaginary unit.

The full method is the quadratic formula itself:

x = (−b ± √(b² − 4ac)) ÷ 2a

The ± symbol means you evaluate it twice — once adding the root and once subtracting — which is why two real roots appear together. The denominator 2a is why a cannot be 0: a zero value would make the equation linear and force a division by zero.

The calculator also reports the vertex, the turning point of the parabola:

vertex x = −b ÷ 2a and vertex y = c − b² ÷ 4a

The x-coordinate of the vertex always sits exactly halfway between the two real roots, on the axis of symmetry.

Terms and units

The coefficients a, b and c are plain numbers and can be positive, negative, whole or decimal — there are no units unless your equation comes from a physical problem. Roots are values of x; the discriminant is a single number; and the vertex is an (x, y) coordinate pair.

Examples

Each example below matches the calculator’s logic exactly. Type the coefficients above to reproduce them.

Example 1: two real roots (a = 1, b = −3, c = −4)

• Discriminant: D = (−3)² − 4(1)(−4) = 9 + 16 = 25. Because D is greater than 0, there are two real roots.
• Roots: x = (3 ± √25) ÷ 2 = (3 ± 5) ÷ 2, giving x = 8 ÷ 2 = 4 and x = −2 ÷ 2 = −1.
• Vertex: x = −(−3) ÷ 2 = 1.5, y = −4 − 9 ÷ 4 = −6.25, so the turning point is (1.5, −6.25).

Example 2: two complex roots (a = 1, b = 2, c = 5)

• Discriminant: D = 2² − 4(1)(5) = 4 − 20 = −16. Because D is less than 0, the roots are complex.
• Roots: x = −2/2 ± (√16 ÷ 2)i = −1 ± (4 ÷ 2)i, giving x = −1 ± 2i.
• The parabola never crosses the x-axis, which is why no real root exists.

Example 3: one repeated root (a = 1, b = −4, c = 4)

• Discriminant: D = (−4)² − 4(1)(4) = 16 − 16 = 0. Because D equals 0, there is one repeated root.
• Root: x = −(−4) ÷ 2 = 4 ÷ 2 = 2, a single value where the curve just touches the x-axis.
• Vertex: x = 2, y = 4 − 16 ÷ 4 = 0, so the vertex (2, 0) sits right on the root.

Discriminant quick-reference table

Discriminant D = b² − 4ac	Number of roots	Root type	Parabola and x-axis
D greater than 0	Two	Two distinct real roots	Crosses at two points
D equal to 0	One (repeated)	One real root	Touches at the vertex
D less than 0	Two	Two complex roots (a ± bi)	Never crosses

What is it used for?

Quadratic equations appear far beyond algebra homework, anywhere a quantity depends on the square of another:

• Projectile motion. The height of a thrown ball over time follows a quadratic, and its roots tell you when it hits the ground.
• Area and dimensions. Sizing a rectangular garden or border of fixed area often reduces to solving ax² + bx + c = 0.
• Optimisation. The vertex gives the maximum or minimum of a quadratic, used for peak profit, lowest cost, or maximum height.
• Physics and engineering. Kinematics, optics, and electrical circuits frequently produce quadratic relationships that need solving.

Tips and common mistakes

• Watch the signs. A negative b becomes positive in −b, and a negative c flips −4ac to a positive addition — exactly what happens in Example 1 where 9 + 16 = 25.
• Compute the discriminant first. Knowing D before the rest tells you instantly whether to expect real or complex roots and prevents you from taking the square root of a negative by mistake.
• Don’t drop the ±. A quadratic usually has two answers; using only the plus sign loses half the solution.
• Keep a nonzero. If your equation has no x² term it is linear — divide through or rearrange instead.
• Verify by substitution. Plug a root back in: for Example 1, 1(4)² − 3(4) − 4 = 16 − 12 − 4 = 0, confirming x = 4 is correct.

Limitations and notes

This tool solves single-variable quadratics only — equations with one x raised to the second power. It will not factor polynomials of higher degree, handle systems of equations, or accept a = 0. Decimal results are rounded for display, so irrational roots (those with a square root that never terminates) are shown as close approximations rather than exact surds; when an exact answer matters, keep the √ form. Complex roots are reported in the standard a ± bi form using the imaginary unit i.

This calculator is an educational and reference tool. For graded coursework, follow the rounding and notation rules your instructor or specification requires.

Related tools

• Pythagorean theorem calculator — solve right triangles, which also rely on squares and square roots.
• Square root calculator — find and simplify the radicals that appear inside the quadratic formula.
• Scientific notation calculator — handle very large or very small coefficients with ease.
• Percentage calculator rounds out everyday arithmetic alongside your algebra work.

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