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Quadratic Formula Calculator

Quadratic Formula Calculator

Quadratic Formula Calculator

Enter the coefficients a, b, and c of a quadratic equation in the form ax² + bx + c = 0. The calculator will find all real and complex roots, the discriminant, vertex, and more.

x = (−b ± √(b² − 4ac)) / 2a
ax² + bx + c = 0
x² +
x +
= 0

What Is a Quadratic Equation?

A quadratic equation is any polynomial equation of degree 2 — meaning the highest power of the variable is 2. It takes the standard form:

ax² + bx + c = 0

where a, b, and c are real numbers called coefficients, and a ≠ 0. If a were equal to zero, the x² term would vanish and the equation would no longer be quadratic — it would become a simple linear equation.

The solutions to a quadratic equation are the values of x that make the equation true. These solutions are called roots, zeros, or x-intercepts — because they are the points where the corresponding parabola crosses the x-axis.

The Quadratic Formula

The quadratic formula is the universal method for solving any quadratic equation. It was derived by completing the square on the general form ax² + bx + c = 0, and it gives the roots directly from the coefficients a, b, and c:

x = (−b ± √(b² − 4ac)) / 2a

The ± symbol means there are two separate solutions — one using addition (x₁) and one using subtraction (x₂). These correspond to the two points where the parabola meets the x-axis.

The Discriminant: b² − 4ac

The expression under the square root — b² − 4ac — is called the discriminant. It is one of the most important values in quadratic equations because it tells you exactly how many real solutions exist before you even finish the calculation.

Δ > 0
Two distinct real roots
The parabola crosses the x-axis at two different points.
Δ = 0
One repeated real root
The parabola just touches the x-axis at one point (the vertex).
Δ < 0
Two complex roots
The parabola does not cross the x-axis at all. Roots involve imaginary numbers.

The Parabola and Its Properties

Every quadratic equation ax² + bx + c = 0 corresponds to a parabola — a U-shaped curve — when graphed as y = ax² + bx + c. Understanding the parabola helps you visualize the roots.

Vertex (h, k) The turning point of the parabola: h = −b/(2a), k = c − b²/(4a)
Axis of symmetry The vertical line x = −b/(2a) that divides the parabola into two mirror halves
Opens upward When a > 0 (minimum vertex — the parabola is a valley)
Opens downward When a < 0 (maximum vertex — the parabola is a hill)
y-intercept The point (0, c) where the parabola crosses the y-axis

Methods for Solving Quadratic Equations

There are four main methods to solve a quadratic equation. Each has its own strengths and is suited to different types of equations.

1. Quadratic Formula — Works for all quadratic equations regardless of whether they factor easily. Always gives exact results, including complex roots.
2. Factoring — The fastest method when it works. Rewrite ax² + bx + c as a product of two binomials. Example: x² − 5x + 6 = (x−2)(x−3) = 0 → x = 2 or x = 3.
3. Completing the Square — Transform the equation into the form (x + p)² = q and then take the square root of both sides. This method also derives the quadratic formula.
4. Square Root Method — Only works when b = 0. If ax² + c = 0, then x = ±√(−c/a). Quick and direct for this special case.

Vieta’s Formulas: Relationship Between Roots and Coefficients

A beautiful and useful property of quadratic equations is that the sum and product of the roots are directly related to the coefficients. These are known as Vieta’s Formulas:

x₁ + x₂ = −b/a The sum of the two roots equals negative b over a
x₁ × x₂ = c/a The product of the two roots equals c over a
Example: x² − 5x + 6 = 0 has roots 2 and 3. Sum = 2+3 = 5 = −(−5)/1 ✓   Product = 2×3 = 6 = 6/1 ✓

Real-World Applications of Quadratic Equations

Quadratic equations appear throughout science, engineering, economics, and everyday life. Whenever a quantity depends on the square of another — such as area, speed, or acceleration — a quadratic equation is likely involved.

Projectile Motion — The height of a thrown ball over time follows h = −½gt² + v₀t + h₀, a quadratic in t. Solving it finds when the ball hits the ground.
Area Problems — Finding the dimensions of a rectangle with a given area and perimeter leads directly to a quadratic equation.
Break-Even Analysis — In economics, profit functions are often quadratic. The roots of the profit equation give the break-even points.
Electrical Engineering — Resonant frequencies in LC circuits are determined by solving quadratic equations involving inductance and capacitance.
Orbital Mechanics — Conic sections (ellipses, parabolas, hyperbolas) all arise from quadratic equations in two variables, describing the paths of planets and satellites.

Complex Roots and Imaginary Numbers

When the discriminant is negative (b² − 4ac < 0), the square root of a negative number appears in the formula. This does not mean there is no solution — it means the solutions are complex numbers involving the imaginary unit i = √(−1).

Complex roots always come in conjugate pairs: if one root is p + qi, the other is always p − qi, where p = −b/(2a) is the real part and q = √(|Δ|)/(2a) is the imaginary part.

i = √(−1)   i² = −1   i³ = −i   i⁴ = 1
Example: x² + 4 = 0 → x = ±√(−4) = ±2i → roots are x₁ = 2i and x₂ = −2i

History of the Quadratic Formula

The quadratic formula has a rich history spanning over 4,000 years. Ancient Babylonian mathematicians (circa 2000 BCE) could solve specific quadratic problems using geometric methods, though they had no symbolic algebra. The Indian mathematician Brahmagupta (628 CE) gave the first explicit general solution for positive roots. The Persian scholar Al-Khwarizmi (820 CE) — whose name gave us the word “algorithm” — systematically classified and solved quadratic equations in his landmark text Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, which also gave us the word “algebra.” The modern symbolic form of the formula was developed in Europe during the 16th and 17th centuries.