Solve second-degree quadratic equations for real and complex roots, discriminant ($b^2 - 4ac$), vertex coordinates, axis of symmetry, focus, directrix, and step-by-step methods.
• Enter coefficients a, b, and c for the standard equation: ax² + bx + c = 0.
• The solver computes exact roots, vertex coordinates, discriminant analysis, and step-by-step algebraic proofs.
Enter coefficients for ax² + bx + c = 0
An in-depth technical resource on algebraic derivations, discriminant classification, and geometric parabola properties.
A quadratic equation is a second-order polynomial equation in a single variable x. The standard algebraic form is:
Here, a is the quadratic coefficient, b is the linear coefficient, and c is the constant term. When graphed on a Cartesian plane, every quadratic equation forms a symmetric U-shaped curve known as a parabola. The points where the parabola crosses the x-axis ($y = 0$) are the roots (also referred to as the solutions or x-intercepts).
The famous Quadratic Formula is derived systematically from ax² + bx + c = 0 by completing the square:
The expression underneath the square root, Δ = b² - 4ac, is called the discriminant. It completely determines the number and nature of roots without solving the entire equation:
Two distinct real roots. The parabola crosses the x-axis at two different locations ($x_1 \ne x_2$).
One repeated real root (multiplicity 2). The parabola touches the x-axis tangentially at its vertex ($x = -b/2a$).
Two complex conjugate roots ($u \pm vi$). The parabola never touches or crosses the x-axis.
The vertex $(h, k)$ represents the turning point or peak/trough of the parabola:
Geometrically, a parabola is the set of all points equidistant from the focus point and the directrix line:
Objects launched under Earth's gravity follow parabolic trajectories described by $h(t) = -\frac12gt^2 + v_0 t + h_0$. Solving the quadratic finds the exact time of impact and maximum height.
In economics, profit functions are commonly modeled as quadratic curves where revenue increases with price until demand drops. The vertex yields the optimal pricing strategy for maximum profit.
Parabolic reflectors reflect incoming parallel light or radio waves directly into a single focal point (the focus), enabling satellite TV dishes, solar concentrators, and telescope mirrors.
No. If a = 0, the equation reduces to bx + c = 0, which is a first-degree linear equation rather than a quadratic. A true quadratic equation must have a non-zero leading coefficient (a ≠ 0).
A negative discriminant means there are no real x-intercepts on the Cartesian plane. Instead, the equation has two complex conjugate roots involving the imaginary unit i (where i = √-1).
The x-coordinate of the vertex is h = -b / (2a). To find the y-coordinate k, substitute h into the original function: k = f(h) = a(h)² + b(h) + c = c - b²/(4a).
Standard form is written as y = ax² + bx + c, which clearly shows the y-intercept (0, c). Vertex form is written as y = a(x - h)² + k, which immediately displays the vertex coordinates (h, k) and axis of symmetry x = h.
Because every positive number has both a positive and negative square root (e.g., √9 = ±3 because 3² = 9 and (-3)² = 9). The ± symbol accounts for both possibilities.