Math

Quadratic Formula Calculator

A comprehensive quadratic formula calculator that solves any equation of the form ax² + bx + c = 0. Get exact and decimal roots, discriminant classification (two real, repeated, or complex), vertex coordinates, axis of symmetry, y-intercept, all three equation forms (standard, vertex, factored), and an interactive parabola graph. Uses a numerically stable algorithm for precise results even with extreme coefficients.

Quick Tips

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Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Standard Form

Classic quadratic x² − 5x + 6 = 0 with two integer roots (x = 2 and x = 3)

Key values: a = 1 · b = −5 · c = 6

Physics Problem

Projectile height equation −4.9t² + 20t + 1.5 = 0 (when does the ball hit the ground?)

Key values: a = −4.9 · b = 20 · c = 1.5

Complex Roots

Equation x² + 2x + 5 = 0 with no real solutions (discriminant < 0)

Key values: a = 1 · b = 2 · c = 5

Documentation

Documentation Contents

How to Use This Calculator

Solve any quadratic equation in three simple steps.

Write your equation in standard form: Rearrange your equation to $ax^2 + bx + c = 0$ .
Enter the coefficients: Type the values of $a$ , $b$ , and $c$ into the input fields. For example, for $2x^2 - 5x + 3 = 0$ , enter a = 2, b = -5, c = 3.
Read the results: The calculator instantly displays the roots (exact and decimal), discriminant, vertex, parabola graph, and all three equation forms.

The calculator handles all cases: two real roots, repeated roots, complex roots, and even linear equations (when a = 0).

The Quadratic Formula

The quadratic formula provides the solutions to any quadratic equation $ax^2 + bx + c = 0$ where $a \neq 0$ :

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Here, $a$ is the coefficient of $x^2$ , $b$ is the coefficient of $x$ , and $c$ is the constant term. The $\pm$ symbol means you evaluate the formula twice: once with addition and once with subtraction, giving two solutions.

Key Properties

The formula works for every quadratic equation, including those that cannot be factored.
The expression under the square root, $b^2 - 4ac$ , is called the discriminant and determines the nature of the roots.
Every quadratic equation has exactly two solutions (counting multiplicity) by the Fundamental Theorem of Algebra.

The Discriminant

The discriminant is defined as:

\Delta = b^2 - 4ac

It determines the nature of the roots before solving:

$\Delta > 0$

Two distinct real roots. The parabola crosses the x-axis at two points.

$\Delta = 0$

One repeated (double) root. The parabola is tangent to the x-axis at the vertex.

$\Delta < 0$

Two complex conjugate roots. The parabola does not cross the x-axis.

When the discriminant is a perfect square, the roots are rational numbers (fractions or integers). When it is positive but not a perfect square, the roots are irrational (involving square roots).

Three Solving Methods

1. The Quadratic Formula

The universal method that works for every quadratic equation. Apply the formula directly:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

2. Factoring

When the discriminant is a perfect square, the equation can be factored. Find two numbers that multiply to $ac$ and add to $b$ , then factor by grouping:

ax^2 + bx + c = a(x - r_1)(x - r_2) = 0

This gives roots $x = r_1$ and $x = r_2$ directly.

3. Completing the Square

Transform the equation into vertex form by creating a perfect square trinomial. This method derives the quadratic formula itself and reveals the vertex form directly. For the step-by-step process, worked examples, and vertex form connection, see the Completing the Square guide.

Worked Examples

Example 1: Projectile Motion

A football is thrown upward from 5 feet at 48 ft/s. The height equation is $h(t) = -16t^2 + 48t + 5$ . When does it hit the ground?

Coefficients: a = -16, b = 48, c = 5

$\Delta = 48^2 - 4(-16)(5) = 2304 + 320 = 2624$

$t = \frac{-48 \pm \sqrt{2624}}{-32} \approx \frac{-48 \pm 51.23}{-32}$

$t_1 \approx -0.10\text{ s (rejected)}, \quad t_2 \approx 3.10\text{ s}$

The ball hits the ground after approximately 3.10 seconds.

Example 2: Garden Dimensions

A garden's length is 3m more than its width, and its area is 108 m². Find the dimensions. Setting up: $w^2 + 3w - 108 = 0$ .

Coefficients: a = 1, b = 3, c = -108

$\Delta = 9 + 432 = 441 = 21^2$ (perfect square)

$w = \frac{-3 \pm 21}{2}$ , giving $w = 9$ or $w = -12$

Width = 9 m, Length = 12 m. Verify: 9 x 12 = 108 m².

Example 3: Complex Roots (No Real Solution)

Does a ball tossed at 20 ft/s from 4 ft ever reach 15 ft? $-16t^2 + 20t - 11 = 0$ .

Coefficients: a = -16, b = 20, c = -11

$\Delta = 400 - 704 = -304$ (negative)

Complex roots: $t = \frac{-20 \pm i\sqrt{304}}{-32} \approx 0.625 \pm 0.545i$

The ball never reaches 15 feet. Maximum height is only 10.25 ft at t = 0.625 s.

Example 4: Repeated Root (Critical Threshold)

Solve $4t^2 - 4t + 1 = 0$ (a ball barely reaching its target height).

Coefficients: a = 4, b = -4, c = 1

$\Delta = 16 - 16 = 0$

$t = \frac{4}{8} = 0.5$ (one repeated root)

The ball touches exactly 10 ft at t = 0.5 seconds, then falls back.

Three Equation Forms

Every quadratic can be expressed in three equivalent forms, each revealing different properties:

Standard Form

ax^2 + bx + c = 0

Best for: identifying coefficients, applying the quadratic formula, general representation.

Vertex Form

a(x - h)^2 + k

Best for: graphing, finding min/max values, identifying the vertex (h, k).

Where $h = -\frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$ .

Factored Form

a(x - r_1)(x - r_2) = 0

Best for: finding roots directly, understanding x-intercepts. Only applies when roots are real.

Vieta's Formulas

Vieta's formulas relate the roots of a quadratic to its coefficients without solving:

r_1 + r_2 = -\frac{b}{a} \qquad r_1 \cdot r_2 = \frac{c}{a}

These relationships hold even for complex roots and provide a quick verification of computed solutions. For example, if $x^2 - 5x + 6 = 0$ has roots 2 and 3, then: 2 + 3 = 5 = -(-5)/1 and 2 x 3 = 6 = 6/1.

Historical Background

The quadratic equation has a 4,000-year history spanning multiple civilizations:

Babylonian origins (~2000-1800 BCE): Clay tablets show scribes solving quadratic problems using geometric methods resembling completing the square, though limited to positive solutions.
Indian mathematicians (5th-12th century CE): Brahmagupta (~628 CE) provided the first explicit algebraic solution acknowledging negative numbers. Bhaskara II (~1150 CE) gave the first clear treatment of the discriminant.
Islamic Golden Age (8th-12th century CE): Al-Khwarizmi (~820 CE) systematically solved all types of quadratic equations with geometric proofs, giving us the word algebra.
European synthesis (12th-17th century CE): The formula reached its modern notation through Descartes (1637) and others. Higher-degree formulas were developed for cubics and quartics, but Abel and Galois proved no general formula exists for degree 5 and above.

Frequently Asked Questions

How do you use the quadratic formula?

Identify coefficients a, b, c from your equation in standard form (ax² + bx + c = 0). Calculate the discriminant b² - 4ac. Substitute into x = (-b ± sqrt(b²-4ac)) / (2a). The ± gives two solutions.

What does a negative discriminant mean?

A negative discriminant means the equation has no real solutions, but two complex conjugate solutions of the form a ± bi. Graphically, the parabola does not cross the x-axis.

Can a quadratic equation have only one solution?

When the discriminant equals zero, the equation has one repeated (double) root. Technically it still has two roots, but they are equal. The parabola touches the x-axis at exactly one point.

What if a = 0?

If a = 0, the equation is not quadratic but linear (bx + c = 0), with solution x = -c/b. The calculator handles this case automatically and displays a clear message.

Why does the quadratic formula work?

The formula is derived by completing the square on the general equation ax² + bx + c = 0. The key steps are: divide by a, move the constant, add (b/2a)² to both sides to create a perfect square, then take the square root and solve for x.

References

Wolfram MathWorld. "Quadratic Equation." Wolfram Research. https://mathworld.wolfram.com/QuadraticEquation.html
Wolfram MathWorld. "Quadratic Formula." Wolfram Research. https://mathworld.wolfram.com/QuadraticFormula.html
Goldberg, D. (1991). What Every Computer Scientist Should Know About Floating-Point Arithmetic. ACM Computing Surveys, 23(1), 5–48.
MacTutor History of Mathematics. "Quadratic, cubic and quartic equations." University of St Andrews. https://mathshistory.st-andrews.ac.uk/HistTopics/Quadratic_etc_equations/
Wikipedia. "Vieta's formulas." https://en.wikipedia.org/wiki/Vieta%27s_formulas

Disclaimer

Educational Tool

This calculator is designed for educational and reference purposes. While it uses a numerically stable algorithm for accurate results, very large coefficients (beyond 10¹⁵) may produce results with reduced decimal precision due to floating-point arithmetic limitations. For mission-critical engineering or scientific calculations requiring arbitrary precision, use specialized mathematical software such as Mathematica, MATLAB, or SageMath.