About This Tool
Solve any quadratic equation of the form ax² + bx + c = 0 by entering the three coefficients. This free calculator applies the quadratic formula to find both roots, determines the discriminant to classify the root type (two real, one repeated, or two complex), computes the vertex coordinates and axis of symmetry, and generates a complete step-by-step solution you can follow or reference for homework. Complex roots are displayed in a + bi format. The equation is shown in formatted mathematical notation as you type, so you can confirm the correct problem before solving. No signup, no installation, and your data stays completely private.
The Quadratic Formula Explained
The quadratic formula solves every equation of the form ax² + bx + c = 0 where a is not zero:
x = (-b ± √(b² - 4ac)) / 2a
This formula was first published in its complete form by the Indian mathematician Brahmagupta in 628 AD, though Babylonian mathematicians had methods for solving specific quadratic equations as early as 2000 BC. The formula works for every quadratic equation without exception, which is why it is taught universally in algebra courses.
The ± symbol means you compute two answers: one using addition and one using subtraction. These two answers are the two roots (solutions) of the equation. Both roots satisfy the original equation, meaning if you substitute either root back in for x, the equation equals zero.
The formula requires three inputs:
- a is the coefficient of x² (must not be zero, otherwise the equation is linear).
- b is the coefficient of x.
- c is the constant term (the number without any x).
Understanding the Discriminant
The expression under the square root sign, b² - 4ac, is called the discriminant (often written as Δ or D). It determines the nature of the roots without actually solving the equation:
- Δ > 0 (positive): The equation has two distinct real roots. The parabola crosses the x-axis at two separate points. Example: x² - 5x + 6 = 0 has Δ = 1, giving roots x = 2 and x = 3.
- Δ = 0 (zero): The equation has exactly one real root (a repeated or double root). The parabola touches the x-axis at exactly one point, the vertex. Example: x² - 4x + 4 = 0 has Δ = 0, giving the single root x = 2.
- Δ < 0 (negative): The equation has no real roots but two complex conjugate roots. The parabola does not cross the x-axis at all. Example: x² + 1 = 0 has Δ = -4, giving roots x = i and x = -i.
The discriminant is useful for quickly checking whether an equation has real solutions before doing the full calculation. In physics and engineering, a negative discriminant often indicates that a system has no real equilibrium point or that two signals never intersect.
Vertex Form and Axis of Symmetry
Every parabola defined by y = ax² + bx + c has a vertex, which is the highest point (if a < 0) or lowest point (if a > 0) on the curve. The vertex coordinates are:
- h = -b / (2a) gives the x-coordinate of the vertex.
- k = c - b² / (4a) gives the y-coordinate of the vertex.
The axis of symmetry is the vertical line x = h that passes through the vertex. The parabola is perfectly symmetric about this line, meaning the two roots are equidistant from it.
Knowing the vertex is essential for graphing. It tells you exactly where the parabola turns, and combined with the direction (up if a > 0, down if a < 0), you can sketch the entire curve. In optimization problems, the vertex represents the maximum or minimum value of the quadratic function.
Example: For y = 2x² - 8x + 3, the vertex is at h = -(-8)/(2*2) = 2, k = 3 - 64/8 = -5. So the vertex is (2, -5), and the parabola opens upward since a = 2 is positive. The minimum value of the function is -5, occurring at x = 2.
Working with Complex Roots
When the discriminant is negative, the square root of a negative number produces an imaginary number. Imaginary numbers use the unit i, where i² = -1. Complex roots always come in conjugate pairs of the form:
- x₁ = a + bi (the "plus" root)
- x₂ = a - bi (the "minus" root, the conjugate)
Here, "a" is the real part and "b" is the imaginary part. Both are calculated from the quadratic formula:
- Real part: -b / (2a)
- Imaginary part: √|Δ| / (2a)
Complex roots have real significance in many fields. In electrical engineering, they describe oscillating circuits. In control systems, they determine whether a system oscillates or remains stable. In signal processing, complex roots of the characteristic equation define the frequency and decay rate of resonance modes.
If you are working on a problem that requires only real answers (like finding where a ball hits the ground), a negative discriminant means the scenario described by the equation has no physical solution. For instance, asking "when does a ball thrown from 10 feet reach 100 feet?" may yield complex roots, meaning the ball never reaches that height.
Frequently Asked Questions
What happens if a = 0?
If a = 0, the equation becomes bx + c = 0, which is a linear equation, not a quadratic equation. Linear equations have at most one solution: x = -c/b (assuming b is also not zero). This calculator requires a nonzero value for "a" because the quadratic formula divides by 2a, which would cause division by zero. If you need to solve a linear equation, simply rearrange bx + c = 0 to x = -c/b.
How do I know which root is x1 and which is x2?
By convention, x₁ uses the "+" sign in the formula (-b + √Δ) / 2a and x₂ uses the "-" sign (-b - √Δ) / 2a. This means x₁ is always greater than or equal to x₂ when the roots are real. The labels are purely for identification; both are equally valid solutions to the equation. You can verify either root by substituting it back into the original equation and confirming the result equals zero.
Can this calculator handle decimal and negative coefficients?
Yes. The calculator accepts any real number for a, b, and c, including decimals (like 2.5), negative values (like -3), and fractions represented as decimals (like 0.333 for 1/3). The only restriction is that "a" cannot be zero. Results are shown with up to 6 decimal places for precision.
What are complex conjugate roots?
Complex conjugate roots are a pair of complex numbers that have the same real part but opposite imaginary parts. If one root is 3 + 2i, the conjugate root is 3 - 2i. Quadratic equations with real coefficients always produce roots that are either both real or complex conjugates of each other. This is guaranteed by the fundamental theorem of algebra and the conjugate root theorem. You will never get one real root and one complex root from a quadratic equation with real coefficients.
How is the vertex used in real-world problems?
The vertex represents the maximum or minimum of a quadratic function, making it central to optimization problems. Projectile motion uses the vertex to find the peak height of a thrown object. Business applications use it to find the price that maximizes revenue (revenue often follows a quadratic model). In engineering, the vertex helps determine the point of minimum stress in an arch or the optimal angle for a reflective dish. Whenever you need to find "the best" or "the most" of something modeled by a parabola, the vertex gives the answer.
What is the difference between roots, zeros, and x-intercepts?
These three terms all refer to the same values but in different contexts. Roots (or solutions) are the values of x that make the equation ax² + bx + c = 0 true. Zeros are the values of x where the function f(x) = ax² + bx + c equals zero. X-intercepts are the points where the graph of y = ax² + bx + c crosses the x-axis (where y = 0). They are all computed the same way using the quadratic formula, factoring, or completing the square.