Quadratic Equation Solver (Formula, Discriminant & Roots)
Solve quadratic equations $ax^2 + bx + c = 0$ with step-by-step discriminant and root solutions.
TL;DR: Quadratic Equation Solver finds the roots of quadratic equations $ax^2 + bx + c = 0$ using the quadratic formula $x = [-b \pm \sqrt{b^2 - 4ac}] / 2a$.
Solves equations in the form aײ + b× + c = 0
What Is the Quadratic Formula and What Does the Discriminant Reveal?
The quadratic formula is: x = [-b ± √(b² - 4ac)] / (2a). The discriminant Δ = b² - 4ac determines root characteristics: if Δ > 0, there are two distinct real roots; if Δ = 0, there is exactly one real repeated root; if Δ < 0, there are two conjugate complex (imaginary) roots.
How to Use the Quadratic Equation Solver
Our Quadratic Equation Solver performs high-precision mathematical operations directly in your browser with zero latency and complete client-side privacy.
- Enter coefficients $a$, $b$, and $c$ from standard form $ax^2 + bx + c = 0$ (where $a \neq 0$).
- Click 'Solve Equation' to calculate the discriminant $\Delta = b^2 - 4ac$.
- Inspect the step-by-step substitution into the quadratic formula.
- View exact root values, decimal approximations, and complex solutions if applicable.
- Review parabola vertex coordinates $(h, k)$ and axis of symmetry.
Mathematical Formula & Equations
Solves second-degree polynomial equations $ax^2 + bx + c = 0$ with discriminant analysis ($\Delta = b^2 - 4ac$) for real and complex roots.
Calculation Example
For $2x^2 + 5x - 3 = 0$: Discriminant = $25 - 4(2)(-3) = 49$. Roots are $x = \frac{-5 \pm 7}{4} \implies x_1 = 0.5, \, x_2 = -3$.
100% Client-Side Privacy & Data Security
All calculations, amortization schedules, variables, and sensitive numerical datasets execute 100% locally in your web browser memory. Your financial, medical, and personal values are never transmitted, logged, or uploaded to any external server.
Frequently Asked Questions
- The standard form is `ax² + bx + c = 0`, where `a`, `b`, and `c` are real numbers and `a ≠ 0`.
- If `b² - 4ac > 0`, the parabola intersects the x-axis twice (two real roots). If `0`, it touches the x-axis once (one real root). If `< 0`, it does not intersect the x-axis (two complex roots).
- The x-coordinate of the vertex is `h = -b / (2a)`. Substitute `h` back into the equation to find the y-coordinate `k = ah² + bh + c`.
- Yes. If the roots are rational, the equation can often be factored into `(px + q)(rx + s) = 0`, where setting each factor to zero yields the roots.
- The axis of symmetry is the vertical line `x = -b / (2a)` that divides the parabola into two mirror-image halves.
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