Solve quadratic equations ax² + bx + c = 0. Get real and complex roots, vertex, discriminant and ste
ax² + bx + c = 0 | x = (−b ± √(b²−4ac)) / 2a
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Root x₁
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Root x₂
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Discriminant (Δ)
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Vertex
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Axis of Symmetry
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Sum of Roots
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Quadratic Equation Guide
A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. It always has exactly two roots (counting multiplicity), which may be real or complex depending on the discriminant Δ = b² − 4ac.
If Δ > 0: two distinct real roots If Δ = 0: one repeated real root If Δ < 0: two complex conjugate roots
Quadratic Equation Solver – Understanding the Quadratic Formula
A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. It is called "quadratic" from the Latin for square — because the highest power of x is 2. The quadratic formula gives exact solutions for any quadratic equation, regardless of whether the roots are real or complex.
The discriminant (Δ = b² − 4ac) reveals the nature of the roots without fully solving the equation. Δ > 0: Two distinct real roots — the parabola crosses the x-axis at two points. Δ = 0: One repeated real root — the parabola just touches the x-axis at its vertex. Δ < 0: Two complex conjugate roots — the parabola does not cross the x-axis at all. Checking the discriminant first helps you understand what type of answer to expect.
The Vertex Form and What It Means Graphically
Every quadratic y = ax² + bx + c corresponds to a parabola. The vertex is the highest or lowest point of the parabola, located at x = −b/(2a), y = c − b²/(4a). The axis of symmetry is the vertical line x = −b/(2a) — the parabola is a perfect mirror reflection across this line. The roots of the equation are where the parabola crosses the x-axis (y = 0). This calculator shows both the algebraic roots and the geometric vertex coordinates.
Real-World Applications of Quadratic Equations
Projectile motion: The height of a thrown ball follows h = −½gt² + v₀t + h₀ (a quadratic in time). Setting h = 0 and solving gives the time of landing. Maximising area: A farmer with 100m of fence enclosing a rectangle against a wall wants to maximise area — this leads to a quadratic optimisation. Break-even analysis: Revenue and cost functions can intersect where their difference = 0, which is often a quadratic. Use our profit & loss calculator for business break-even calculations.
A quadratic equation is a polynomial equation of degree 2 in the form ax² + bx + c = 0, where a is not zero. It always has exactly two roots (solutions), which may be two distinct real numbers, one repeated real number, or two complex numbers, depending on the discriminant.
What is the quadratic formula?+
The quadratic formula gives the solutions to any quadratic equation: x = (-b ± square root of (b² - 4ac)) / (2a). The ± means there are two solutions: one using addition and one using subtraction. This formula always works regardless of whether the roots are real or complex.
What is the discriminant?+
The discriminant is b² - 4ac, the part under the square root in the quadratic formula. If it is positive, there are two distinct real roots. If it equals zero, there is one repeated real root (the parabola just touches the x-axis). If it is negative, there are two complex (non-real) roots and the parabola does not cross the x-axis.
What do the roots of a quadratic represent graphically?+
The roots are the x-intercepts of the parabola y = ax² + bx + c — where the curve crosses the x-axis. Two real roots mean two crossing points. One repeated root means the vertex touches the x-axis. No real roots mean the parabola is entirely above or below the x-axis.
What are real-world applications of quadratic equations?+
Quadratic equations model projectile motion (the path of a thrown ball), area calculations (maximising a rectangular enclosure), profit optimisation in business (where revenue minus cost is a quadratic), physics (falling objects), engineering (beam deflection), and finance (calculating compound interest over specific periods).