Quadratic Equation Solver
Solve quadratic equations ax² + bx + c = 0 with comprehensive step-by-step math breakdowns, behavior presets, and real-time interactive plotting.
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Detailed Mathematical Solution
Equation coefficients: a = 3, b = 5, c = -2
Discriminant Δ = b² - 4ac = (5)² - 4(3)(-2)
Δ = 25 - (-24) = 49
Since Δ > 0, equation has two distinct real roots: x = (-b ± √Δ) / 2a
x = (-(5) ± √49) / (2 * 3)
x = (-5 ± 7.0000) / 6
x₁ = (-5 + 7.0000) / 6 = 0.3333 (1/3)
x₂ = (-5 - 7.0000) / 6 = -2.0000
Understanding Quadratic Equations
A quadratic equation is a second-order algebraic polynomial. The graph of a quadratic equation represents a symmetrical curve known as a parabola.
Standard Quadratic Formula
x = [ -b ± √(b² - 4ac) ] / 2a
What Vertex represents?
The vertex coordinate represents the absolute minimum point (if opens up, where a > 0) or maximum point (if opens down, where a < 0) of the parabola.
Common FAQs
What if the discriminant (Δ) is negative?
How is the Axis of Symmetry calculated?
What does y-intercept represent?
ax² + bx + c, Solved and Explained
The discriminant tells you before you solve
b² − 4ac decides what kind of answer exists:
| Discriminant | Roots | Graph meaning |
|---|---|---|
| Positive | Two distinct real roots | Parabola crosses the x-axis twice |
| Zero | One repeated root | Parabola touches the axis once |
| Negative | Two complex roots | Parabola never reaches the axis |
| Perfect square (and rational a,b,c) | Rational roots | It would have factored nicely |
Three methods, one decision rule
Factoring is fastest when it works (integer-friendly roots), completing the square is what derives the vertex form (and the formula itself), and the quadratic formula always works — it's the safety net. Exam strategy: try ten seconds of factoring; if nothing obvious appears, go straight to the formula and bank the time. The solver shows all steps, which is what matric/FSc marking schemes actually award.
Quadratics outside the textbook
Projectile motion (height vs time is a parabola), profit maximization, area-with-fixed-perimeter problems, and the px²-shaped cost curves of physics and economics homework. Geometry-flavored quadratics — right triangles via Pythagoras — get a dedicated treatment in the triangle calculator, and everything else evaluates in the scientific calculator.