Controls width and direction. Positive a opens upward ↑; negative opens downward ↓. Larger |a| means a narrower, steeper parabola.
b
Coefficient b
Shifts the axis of symmetry. The axis is always at x = −b/(2a), so changing b moves the vertex left or right.
c
Coefficient c
Shifts the entire parabola up or down without changing its shape. c is exactly the y-intercept: the parabola always passes through (0, c).
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How to Use This Calculator
1
Enter a, b, and c
Type the three coefficients into the labeled inputs. Use the preset equation buttons for quick examples. a must not be zero — the calculator flags it if so.
2
Read the Results
The discriminant badge shows root type (real, repeated, or complex). Both roots appear below in simplified form. Sub-stats show vertex coordinates, axis of symmetry, and y-intercept.
3
View the Parabola
The chart at the bottom of the results card draws the parabola with roots, vertex, and axis of symmetry labeled. Toggle "Show Step-by-Step Work" to see all 5 solution steps with your values substituted.
4
Learn with the Lesson Tab
Switch to the Step-by-Step Lesson tab and click Next to walk through the derivation personalized to your equation. Use the Parabola Explorer tab to see how each coefficient changes the curve.
DiscriminantThe value b²−4ac that determines the number and type of roots. Positive → two real roots, zero → one repeated root, negative → two complex roots.
Roots / ZerosThe x values that satisfy ax²+bx+c=0. Geometrically, they are the x-intercepts of the parabola (when real).
VertexThe turning point of the parabola. It is the minimum when a>0 (opens up) and the maximum when a<0 (opens down).
Axis of SymmetryThe vertical line x = −b/(2a) that passes through the vertex and splits the parabola into two mirror images.
ParabolaThe U-shaped curve described by a quadratic function. It is symmetric about its axis of symmetry and has one vertex.
Complex NumberA number of the form a + bi where i = √(−1). Complex roots always appear in conjugate pairs (a+bi and a−bi) for equations with real coefficients.
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Worked Examples
x²
x² − 5x + 6 = 0
a = 1, b = −5, c = 6
Discriminant
Δ = 25 − 24 = 1
x₁ = (5 + 1) / 2 = 3 and x₂ = (5 − 1) / 2 = 2. Vertex at (2.5, −0.25). The parabola crosses the x-axis at x = 2 and x = 3.
x²
x² − 2x + 1 = 0
a = 1, b = −2, c = 1
Discriminant
Δ = 4 − 4 = 0
x = 2 / 2 = 1 (one repeated root). This is a perfect square trinomial: (x−1)² = 0. The parabola touches the x-axis at exactly one point: its vertex (1, 0).
x²
x² + x + 1 = 0
a = 1, b = 1, c = 1
Discriminant
Δ = 1 − 4 = −3
x = −1/2 ± (√3/2)i. Since Δ < 0 the parabola has no real x-intercepts — it floats entirely above the x-axis. The complex conjugate roots are ≈ −0.5 ± 0.866i.
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Quadratic Equations in the Real World
Projectile Motion
When you throw a ball upward, its height h(t) at time t follows a quadratic equation: h = −½gt² + v₀t + h₀, where g is gravitational acceleration, v₀ is initial velocity, and h₀ is starting height. Solving h = 0 gives the times when the ball hits the ground — a direct application of the quadratic formula. Engineers use this in ballistics, sports science, and rocket trajectory planning.
Business Profit Optimization
Revenue and cost functions in economics are often quadratic. For example, if a company sets price p and the demand function is Q = 100 − 2p, then revenue R = p·Q = 100p − 2p². To find the profit-maximizing price, you take the derivative and set it to zero — which is equivalent to finding the vertex of the parabola. The quadratic formula is also used to find break-even points where profit = 0.
Physics and Engineering
Quadratic equations appear in Ohm's law (power dissipation P = I²R), lens formulas, and the equations of motion for objects under constant force. In structural engineering, the shape of arches and suspension cables follows parabolic equations, and load calculations require solving quadratic inequalities to ensure safety margins are met.
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Frequently Asked Questions
What is the quadratic formula?+
x = (−b ± √(b²−4ac)) / (2a). It solves any equation of the form ax²+bx+c=0 for the values of x. The ± means there are two solutions: one with + and one with −.
What is the discriminant and what does it tell me?+
Δ = b²−4ac. If Δ > 0 there are two distinct real roots (the parabola crosses the x-axis twice). If Δ = 0 there is one repeated root (the parabola just touches the x-axis). If Δ < 0 there are two complex roots (no real x-intercepts).
What are complex roots and how do I interpret them?+
Complex roots take the form a ± bi where i = √(−1). They mean the parabola never crosses the x-axis — it is entirely above (if a > 0) or below (if a < 0) the x-axis. Complex roots always appear in conjugate pairs for equations with real coefficients.
How is the vertex calculated?+
The vertex x-coordinate is h = −b/(2a). Substitute back: k = a·h²+b·h+c. This gives the turning point (h, k). When a > 0 the vertex is the minimum; when a < 0 it's the maximum.
When should I use the quadratic formula vs factoring?+
Factoring is faster when integer roots exist (discriminant is a perfect square, small coefficients). Use the quadratic formula for decimal coefficients, large numbers, or when you need complex roots. The formula always works regardless of root type.
Why does a have to be non-zero?+
If a = 0 the equation becomes bx+c = 0 — a linear equation, not quadratic. The term ax² disappears and the quadratic formula involves dividing by 2a = 0, which is undefined. Linear equations have at most one solution and are solved differently.
What is vertex form and how do I convert to it?+
Vertex form is a(x−h)²+k. Compute h = −b/(2a) and k = a·h²+b·h+c, then substitute. For example, x²−4x+3 → h = 2, k = −1, so vertex form is (x−2)²−1.
What does the chart show?+
The parabola chart marks real roots as cyan dots on the x-axis, the vertex as a gold dot, and the y-intercept as a teal dot. The dashed vertical line is the axis of symmetry at x = −b/(2a). For equations with complex roots the parabola does not touch the x-axis.
Can I solve equations with non-integer coefficients?+
Yes. Enter decimals directly (e.g. a = 1.5, b = −2.7, c = 0.3). The calculator handles any real-number coefficients and displays roots as simplified fractions when possible, or as decimals otherwise.
What is the axis of symmetry?+
The vertical line x = −b/(2a) that splits the parabola into two mirror images. Every parabola has exactly one axis of symmetry, and it always passes through the vertex.