Solve 1x² - 3x + 2 = 0
The real or complex roots of this quadratic equation, using the quadratic formula. Adjust any coefficient below to solve your own equation.
Roots
x₁ = 2, x₂ = 1
For x² - 3x + 2 = 0, the discriminant is 1, giving two real roots: x = 2 and x = 1.
Discriminant
1
Vertex
(1.5, -0.25)
Parabola Opens
Upward
Real Roots?
Yes
What Is a Quadratic Equation?
A quadratic equation is any second-degree polynomial equation of the form ax² + bx + c = 0, where a cannot equal zero (otherwise it reduces to a linear equation). Its solutions — called roots — are the x-values where the corresponding parabola y = ax² + bx + c crosses the x-axis.
Quadratic equations show up constantly in real-world modeling: projectile motion, area optimization problems, and any relationship where a quantity depends on the square of another variable.
Plot of y = ax² + bx + c
The Quadratic Formula
Any quadratic equation can be solved directly using the quadratic formula, derived by completing the square on the general form ax² + bx + c = 0.
The Discriminant Determines the Type of Roots
The expression under the square root, b² − 4ac, is called the discriminant. When it's positive, the equation has two distinct real roots. When it equals zero, there's exactly one repeated real root (the parabola's vertex touches the x-axis). When it's negative, the roots are a complex-conjugate pair, and the parabola never crosses the x-axis at all.
The Vertex and Axis of Symmetry
Every parabola is symmetric around a vertical line called the axis of symmetry, located at x = −b ÷ 2a — which is also the x-coordinate of the vertex, the parabola's minimum point (if a > 0) or maximum point (if a < 0).
Real-World Applications
Projectile motion under gravity follows a quadratic path, so the quadratic formula finds exactly when a thrown object hits the ground. Quadratics also solve optimization problems like maximizing a rectangular area for a fixed perimeter, and appear throughout physics, engineering, and finance wherever a squared relationship applies.
Example — Your Current Inputs
For x² - 3x + 2 = 0, the discriminant is 1, giving two real roots: x = 2 and x = 1.
Additional Example — A Ball Thrown Upward
A ball thrown upward follows height h = −16t² + 64t + 5 (feet, seconds). Setting h = 0 and solving −16t² + 64t + 5 = 0 with the quadratic formula gives t ≈ 4.08 seconds — the moment the ball hits the ground, ignoring the negative time root, which isn't physically meaningful.
About These Parameters
- Coefficient a
- The coefficient of x², controlling whether the parabola opens upward (a > 0) or downward (a < 0), and how narrow or wide it is. Cannot be zero.
- Coefficient b
- The coefficient of the linear x term, which shifts the parabola's vertex left or right and affects the axis of symmetry.
- Coefficient c
- The constant term — also the parabola's y-intercept, the value of y when x = 0.
Frequently Asked Questions
What does it mean when the roots are complex?
A negative discriminant means the parabola never crosses the x-axis — there's no real number solution. The roots still exist mathematically as a complex-conjugate pair, useful in fields like electrical engineering, but they don't correspond to points on a standard x-y graph.
Can I use fractions as coefficients?
Yes — enter a decimal equivalent (like 0.75 for 3/4) in any of the coefficient fields, and the calculator solves the equation exactly the same way.
What if a is zero?
If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0), which has at most one solution and isn't a quadratic equation anymore — this calculator requires a nonzero value for a.