What is the quadratic formula?

The quadratic formula is a universal method for solving any quadratic equation of the form ax² + bx + c = 0 (where a ≠ 0): x = (−b ± √(b² − 4ac)) / (2a). It gives the x-values (roots) at which the parabola y = ax² + bx + c crosses the x-axis. The formula was known to ancient Babylonian and Indian mathematicians and was formally published in the general form in Europe in the 16th century.

How do you solve a quadratic equation using the quadratic formula?

Identify the coefficients a, b, and c from your equation. Compute the discriminant: Δ = b² − 4ac. Then substitute into x = (−b ± √Δ) / (2a). If Δ > 0, you get two distinct real roots; if Δ = 0, one repeated root; if Δ < 0, two complex roots. Example: 2x² − 5x + 2 = 0 → a=2, b=−5, c=2 → Δ = 25 − 16 = 9 → x = (5 ± 3) / 4 → x₁ = 2.00, x₂ = 0.50.

What is the discriminant and what does it tell you?

The discriminant is Δ = b² − 4ac, the expression under the square root in the quadratic formula. It tells you the nature of the roots without fully solving the equation. Δ > 0: two distinct real roots (the parabola crosses the x-axis twice). Δ = 0: one repeated real root (the parabola is tangent to the x-axis). Δ < 0: no real roots — the parabola never crosses the x-axis — but there are two complex conjugate roots.

What does it mean when the discriminant is negative?

A negative discriminant means the quadratic equation has no real-number solutions. The parabola lies entirely above or entirely below the x-axis. The solutions exist as complex numbers: x = (−b ± i√|Δ|) / (2a), where i = √(−1). Complex roots always come in conjugate pairs — if x₁ = p + qi, then x₂ = p − qi. This appears in electrical engineering (AC circuits), signal processing, and control systems.

What is the vertex of a parabola and how do you find it?

The vertex is the highest or lowest point on the parabola y = ax² + bx + c. Its x-coordinate is x_v = −b/(2a), and its y-coordinate is found by substituting back: y_v = a(x_v)² + b(x_v) + c. If a > 0, the vertex is the minimum; if a < 0, it is the maximum. The vertex is equidistant from both roots, so x_v is always the average of x₁ and x₂.

Can every quadratic equation be solved with the quadratic formula?

Yes — the quadratic formula always works for any quadratic equation with real coefficients, regardless of whether the roots are integers, fractions, irrational numbers, or complex numbers. However, other methods may be faster in special cases: factoring works when the roots are rational and easy to spot; completing the square is useful for deriving the vertex form; and for equations like x² = k, simply taking the square root is faster. The quadratic formula is the guaranteed fallback when other methods are difficult to apply.

What are real-world examples of quadratic equations?

Quadratics appear everywhere: projectile motion (h = −16t² + v₀t + h₀ models the height of a ball thrown upward), revenue optimization (profit = price × quantity − cost, where both change), finding the dimensions of a rectangle with a given area and perimeter, and electrical engineering (AC circuit impedance). Structural engineers use quadratics to calculate deflection in beams. Any relationship that involves a squared term produces a quadratic.

What does it mean when a quadratic has no real solutions?

When the discriminant b² − 4ac is negative, the parabola y = ax² + bx + c never touches the x-axis — it is entirely above (if a > 0) or entirely below (if a < 0) the x-axis. In real-world terms: a ball that you throw upward does not reach a certain height (discriminant negative → no time when the ball is at that height). The solutions exist as complex numbers (involving i = √−1), which appear in electrical engineering for AC phase analysis but are not physically realizable as real measurements.

Quadratic Formula Calculator — Solve ax² + bx + c

Solves any quadratic equation ax² + bx + c = 0 — shows the discriminant, roots, factored form, and vertex of the parabola.

How to Use the Quadratic Formula Calculator

This quadratic formula calculator solves any equation of the form ax² + bx + c = 0 instantly — enter the three coefficients a (the coefficient of x²), b (the coefficient of x), and c (the constant term), and it shows the discriminant, the nature of the roots, both root values, the factored form of the equation, and the vertex of the corresponding parabola. If the discriminant is negative, complex roots are displayed in standard form (p ± qi).

The coefficient a cannot be zero — if it were, the equation would be linear (bx + c = 0) rather than quadratic. For relationships between variables and ratio problems, see our proportion calculator.

The Quadratic Formula

Every quadratic equation ax² + bx + c = 0 (with a ≠ 0) can be solved by substituting its coefficients into the quadratic formula:

x = (−b ± √(b² − 4ac)) / (2a)

The ± symbol means you compute two values: one with addition and one with subtraction. These are the two roots x₁ and x₂. The expression under the radical, Δ = b² − 4ac, is the discriminant — it determines whether you get two real roots, one repeated root, or two complex roots.

Worked Example: Two Real Roots

Solve x² − 5x + 6 = 0 (a = 1, b = −5, c = 6):

Discriminant: Δ = (−5)² − 4(1)(6) = 25 − 24 = 1
Δ > 0 → two distinct real roots
x₁ = (5 + 1) / 2 = 3.00
x₂ = (5 − 1) / 2 = 2.00
Factored form: (x − 3)(x − 2) = 0
Vertex: x_v = 5/2 = 2.50, y_v = (2.50)² − 5(2.50) + 6 = −0.25

Worked Example: Complex Roots

Solve x² + 2x + 5 = 0 (a = 1, b = 2, c = 5):

Discriminant: Δ = 4 − 20 = −16
Δ < 0 → no real roots; roots are complex
x = (−2 ± √(−16)) / 2 = (−2 ± 4i) / 2
x₁ = −1 + 2i, x₂ = −1 − 2i

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Understanding the Discriminant

The discriminant Δ = b² − 4ac is the expression under the square root. Its value determines the nature and count of solutions before you even finish the calculation:

Δ > 0 — two distinct real roots. The parabola crosses the x-axis at two different points. The larger Δ is, the farther apart the roots. For example, Δ = 100 gives roots that are 10 apart (along a unit-a equation).
Δ = 0 — one repeated real root (a double root) at x = −b/(2a). The parabola is tangent to the x-axis — it touches but does not cross. The factored form has a squared factor: a(x − r)².
Δ < 0 — no real roots. The parabola does not intersect the x-axis. Solutions are complex conjugates: x = (−b ± i√|Δ|) / (2a). These appear naturally in oscillatory systems such as spring-mass-damper models and RLC circuits.

Knowing the discriminant also tells you about the factorability of the quadratic: if Δ is a perfect square, the equation factors over the integers. If Δ = 1 (as in the first example above), the roots are integers or simple fractions.

The Vertex of the Parabola

Every quadratic equation y = ax² + bx + c defines a parabola. The vertex is its extreme point — either the minimum (when a > 0) or the maximum (when a < 0):

Vertex x-coordinate: x_v = −b / (2a)
Vertex y-coordinate: y_v = a(x_v)² + b(x_v) + c

The vertex x-coordinate is the axis of symmetry of the parabola — the roots (when they exist) are always symmetric around this value. For example, if x₁ = 2 and x₂ = 6, the vertex is at x_v = 4. The vertex is critical in optimization problems: the maximum or minimum value of a quadratic expression always occurs there. For slope and line problems in coordinate geometry, our slope calculator handles the linear side.

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Methods for Solving Quadratic Equations

The quadratic formula always works, but other methods may be faster in specific cases:

Factoring — works when the roots are rational and easy to identify. For x² − 5x + 6 = 0, recognize that −3 × −2 = 6 and −3 + −2 = −5, so (x−3)(x−2) = 0. Fastest when it applies, but requires pattern recognition.
Square root method — fastest when b = 0. For 3x² − 12 = 0 → x² = 4 → x = ±2. No formula needed.
Completing the square — convert to vertex form ax² + bx + c = a(x + b/2a)² − (b²−4ac)/4a. Useful for deriving the vertex and for deriving the quadratic formula itself.
Quadratic formula — always works, no pattern recognition required, handles complex roots automatically. The best choice when factoring is not obvious or when you need a decimal answer quickly.

For most homework and real-world problems, the quadratic formula is the safest choice — it never fails and requires no creativity beyond substituting three numbers. When working with square roots that appear in the quadratic formula, our radical calculator simplifies any radical expression or converts it to a decimal. To find the inverse of the parabola function (restricting domain as needed), see our inverse function calculator.

Sources & References

Quadratic Formula — Algebra Reference — Khan Academy
Quadratic Equation — Wolfram MathWorld — Wolfram Research

Quadratic Formula Calculator