Quadratic Function Calculator
Enter coefficients for the equation ax² + bx + c = 0 to calculate roots, discriminant, vertex, and axis of symmetry.
Tip: Try a = 1, b = -3, c = 2 to get two real roots.
What Is a Quadratic Function?
A quadratic function is a polynomial of degree 2, usually written as f(x) = ax² + bx + c. The graph is a parabola that opens upward when a > 0 and downward when a < 0. Quadratics appear everywhere: projectile motion, area optimization, break-even analysis, and many physics and engineering models.
This calculator helps you quickly solve the equation and interpret the shape of the parabola without doing all the algebra by hand.
What This Calculator Gives You
- Discriminant (D = b² - 4ac): tells how many real solutions exist.
- Roots (x-intercepts): real or complex solutions of ax² + bx + c = 0.
- Vertex: turning point of the parabola.
- Axis of symmetry: vertical line through the vertex.
- Y-intercept: point where the graph crosses the y-axis.
How the Math Works
1) Discriminant Test
The value of D = b² - 4ac determines root type:
- D > 0: two distinct real roots
- D = 0: one repeated real root
- D < 0: two complex conjugate roots
2) Quadratic Formula
When a ≠ 0, roots are found with:
x = (-b ± √(b² - 4ac)) / (2a)
3) Vertex and Axis
The x-coordinate of the vertex is h = -b/(2a). Then compute k = f(h). So the vertex is (h, k) and the axis is x = h.
Example Walkthrough
Suppose you enter a = 1, b = -3, c = 2. The equation is x² - 3x + 2 = 0.
- Discriminant: D = 9 - 8 = 1
- Roots: x = (3 ± 1)/2 → x = 2 and x = 1
- Vertex: h = 3/2 = 1.5, k = -0.25
- Parabola opens upward because a is positive
So the parabola crosses the x-axis at 1 and 2, and has a minimum at (1.5, -0.25).
Common Mistakes to Avoid
- Forgetting that a cannot be 0 if you want a quadratic.
- Dropping parentheses when evaluating -b or 2a.
- Mixing up signs in b² - 4ac.
- Assuming no real roots means “no roots” (complex roots still exist).
When to Use a Quadratic Calculator
Use it when you need quick, reliable results while studying algebra, checking homework, building graphing intuition, or solving applied problems in science and finance. It is also useful for validating manual work and finding errors early.
Final Notes
A good quadratic calculator does more than output two numbers. It should help you interpret the equation and connect algebra to the graph. Use the tool above, then verify one result manually to reinforce the concept.