Solve ax²+bx+c=0 | Discriminant | Vertex | Step-by-Step Solutions
✓ 2 Methods - Formula & completing square
✓ Discriminant - Nature of roots analysis
✓ Complex Roots - Full support
✓ Vertex - Maximum/minimum point
Solve ax² + bx + c = 0 | Two Different Methods | Step-by-Step Solutions
Quadratic Formula: Direct substitution method - fastest way to get roots!
Completing the Square: Algebraic manipulation - shows how to transform equation into vertex form!
Coefficient of x² (a ≠ 0)
Coefficient of x
Constant term
No history yet
📐 Quadratic Formula
✓ Direct substitution into x = (-b ± √Δ)/(2a)
✓ Fastest method
✓ Works for all cases
✏️ Completing Square
✓ Algebraic manipulation
✓ Shows vertex form (x-h)²=k
✓ Educational approach
Choose between quadratic formula for quick solutions or completing the square for deeper understanding. Both methods show complete step-by-step working.
Automatically calculates discriminant (Δ=b²-4ac) and determines nature of roots including real, equal, or complex conjugate roots with full explanation.
Calculates vertex coordinates, identifies parabola direction, and determines maximum or minimum point for complete quadratic function analysis.
A Quadratic Equation Calculator is specialized mathematical tool solving second-degree polynomial equations form ax² plus bx plus c equals zero where a not equal zero essential algebra tool throughout mathematics science engineering solving equations appearing projectile motion profit optimization area calculations physics problems quadratic equations representing parabolic relationships fundamental algebra standard form ax²+bx+c equals zero where a coefficient x-squared leading coefficient determining parabola opens upward positive downward negative b coefficient x affecting axis symmetry vertex location c constant term y-intercept affecting vertical position equation solving methods quadratic formula most common direct method x equals negative b plus minus square root b-squared minus 4ac all divided 2a working every quadratic equation providing exact solutions both real complex roots memorizable formula quick calculations completing square algebraic manipulation converting standard form vertex form a times x minus h squared plus k revealing vertex h k directly showing geometric properties parabola educational method understanding structure deriving quadratic formula itself factoring alternative method when equation factors easily finding roots setting each factor zero fastest when recognizable patterns graphical method plotting parabola finding x-intercepts visual understanding approximate solutions numerical methods iterative approximations large complex systems making calculator essential providing multiple solution methods step-by-step explanations discriminant analysis.
Our Free Quadratic Equation Calculator 2025 offers comprehensive features exceptional functionality two solution methods quadratic formula fastest most straightforward applying memorized formula directly obtaining roots suitable quick solutions exam situations homework problems completing square more educational revealing vertex form showing parabola structure understanding geometric interpretation deriving fundamental relationships calculator supporting both methods educational flexibility discriminant analysis crucial component calculator automatically computing discriminant delta equals b-squared minus 4ac determining nature roots three possibilities discriminant positive two distinct real roots parabola crossing x-axis two points roots different real numbers discriminant zero one repeated real root parabola touching x-axis exactly one point vertex both roots identical equal discriminant negative two complex conjugate roots parabola not intersecting x-axis roots complex numbers form a plus bi calculator handling all cases showing appropriate root types vertex calculation computing vertex coordinates h equals negative b divided 2a k obtained substituting h original equation vertex representing turning point maximum when a negative minimum when a positive crucial optimization problems finding extreme values parabola direction identification detecting whether parabola opens upward a positive downward a negative affecting interpretation vertex maximum minimum applications complex roots support when discriminant negative calculator computing complex conjugate roots real imaginary parts displaying format a plus minus bi explaining conjugate relationship important advanced mathematics engineering applications step-by-step solutions detailed working every calculation showing formula application intermediate steps final roots educational transparent helping students understand process learn solution methods verify manual calculations calculation history tracking last ten solved equations reviewing previous work comparing solutions learning patterns mobile responsive design adapting all screen sizes smartphones tablets desktops ensuring accessibility anywhere anytime making most powerful comprehensive user-friendly quadratic calculator available online completely free unlimited use supporting mathematics education worldwide.
| Equation | Coefficients | Discriminant | Nature | Roots |
|---|---|---|---|---|
| x² - 5x + 6 = 0 | a=1, b=-5, c=6 | Δ = 1 | Two real roots | x = 2, 3 |
| x² - 4x + 4 = 0 | a=1, b=-4, c=4 | Δ = 0 | Equal roots | x = 2 (repeated) |
| x² + x + 1 = 0 | a=1, b=1, c=1 | Δ = -3 | Complex roots | x = -0.5 ± 0.866i |
| 2x² - 8x + 6 = 0 | a=2, b=-8, c=6 | Δ = 16 | Two real roots | x = 1, 3 |
Enter coefficients a, b, c from ax²+bx+c=0. Select method (Quadratic Formula or Completing Square). Click Solve. Calculator shows: discriminant (Δ=b²-4ac), nature of roots, two roots (x₁, x₂), vertex (h,k), parabola direction, complete step-by-step solution. Example: x²-5x+6=0 → a=1, b=-5, c=6 → Δ=1, roots x=2,3. Perfect for homework, exams, learning.
Discriminant Δ=b²-4ac determines root types: Δ>0: Two distinct real roots, parabola crosses x-axis twice. Δ=0: One repeated root, parabola touches x-axis once. Δ<0: Two complex roots, parabola doesn't cross x-axis. Calculator automatically calculates, interprets, and shows geometric meaning. Example: x²+x+1=0 has Δ=-3<0 → complex roots.
Steps: 1) Divide by a if a≠1. 2) Move constant right. 3) Add (b/2a)² both sides. 4) Factor as perfect square (x+p)²=q. 5) Take square root. 6) Solve for x. Calculator shows each algebraic step, perfect square formation, square root process, final roots. Example: x²-6x+5=0 → (x-3)²=4 → x=5 or x=1. Educational method showing vertex form.
Vertex is turning point: highest (a<0) or lowest (a>0). Formula: h=-b/(2a), k=f(h). Example: x²-4x+7=0 → h=2, k=3, vertex (2,3). If a>0: opens upward, minimum at vertex. If a<0: opens downward, maximum at vertex. Calculator provides coordinates, type (min/max), direction. Used for optimization problems, projectile motion, graphing.
When Δ<0: x=(-b±i√|Δ|)/(2a). Real part: -b/(2a). Imaginary part: ±√|Δ|/(2a). Complex conjugate pair: x₁=α+βi, x₂=α-βi. Example: x²+2x+5=0 → Δ=-16 → x=-1±2i. Calculator shows: discriminant indicating complex, real/imaginary parts, both conjugate roots, step-by-step. Parabola never crosses x-axis. Used in engineering, physics.
FORMULA: Direct x=(-b±√Δ)/(2a), fastest, good for quick solutions. SQUARE: Algebraic steps forming (x+p)²=q, shows vertex form, educational. Both give same roots. Formula faster for answers, Square better for understanding concepts and finding vertex directly. Calculator supports both for flexibility, learning, verification.
Δ=0 means one repeated root: x=-b/(2a). Both roots identical. Parabola touches x-axis at one point (vertex). Perfect square: a(x-r)²=0. Example: x²-6x+9=0 → Δ=0 → x=3 (repeated) → (x-3)²=0. Calculator shows: Δ=0, single root, vertex at root. Used for optimization (boundary conditions), critical damping, tangent motion.
a<0: Parabola opens downward ⬇, vertex is maximum (highest point). Calculations same with negative values. Example: -x²+4x-3=0 → a=-1, b=4, c=-3 → x=1,3, vertex (2,1) maximum. Calculator correctly: detects negative a, shows Downward direction, labels vertex as Maximum, handles sign arithmetic. Used for profit maximization, projectile height.
Yes! After setting up equation: 1) Read problem, identify unknowns. 2) Form equation ax²+bx+c=0. 3) Enter coefficients. 4) Solve. 5) Interpret in context. Types: area/geometry, projectile motion (h=-16t²+v₀t+h₀), business (profit), number problems. Calculator helps solve after setup. Example: Ball thrown at 48ft/s from 6ft. When hits ground? -16t²+48t+6=0 → t≈3.12s.
100% free forever! All features: Two methods (formula, completing square), complete step-by-step, discriminant analysis, nature of roots, real/equal/complex roots support, vertex calculation, parabola direction, history (10 items), high precision (4 decimals), mobile responsive. No registration, payment, ads, downloads, limits. Works any device/browser. Perfect for students, teachers, homework, exams, learning algebra. Free for everyone!
Our Free Quadratic Equation Calculator 2025 is the most advanced comprehensive quadratic equation solving tool supporting standard form ax²+bx+c=0 providing two professional solution methods quadratic formula completing square complete step-by-step solutions discriminant analysis nature roots determination vertex calculation parabola direction identification complex roots support calculation history high precision results completely free unlimited use no registration advertisements serving students teachers educators anyone needing reliable accurate quadratic equation solver updated 2025 standards modern educational practices mathematical excellence.