Quadratic Equation Solver
Fast, accurate and free online rozwiazywanie rownan kwadratowych tool running directly in your browser.
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Other tools you may find usefulSolving quadratic equations - delta and roots with steps
The quadratic equation (ax² + bx + c = 0) is one of the basic problems of algebra. The calculator calculates the discriminant (delta = b² - 4ac), real or complex roots and the product form, showing all solution steps.
Quadratic equation formulas
ax² + bx + c = 0 (a ≠ 0). Delta: Δ = b² - 4ac. If Δ > 0: two real roots x1,2 = (-b ± √Δ) / (2a). If Δ = 0: one double root x = -b/(2a). If Δ < 0: no real roots (two complex x1,2 = (-b ± i√|Δ|) / (2a)).
Product form
ax² + bx + c = a(x - x1)(x - x2). Check: expand → return to general form. Viète's formulas: x1 + x2 = -b/a. x1 × x2 = c/a. Symmetry: a parabola has an axis of symmetry x = -b/(2a) = (x1+x2)/2. Vertex: x_v = -b/(2a), y_v = -Δ/(4a).
Calculation examples
x² - 5x + 6 = 0: a=1, b=-5, c=6. Δ=25-24=1. x1=(5+1)/2=3, x2=(5-1)/2=2. Check: (x-3)(x-2)=x²-5x+6 ✓. x² + 2x + 5 = 0: Δ=4-20=-16 < 0. No real roots. Complex elements: x = (-2 ± 4i)/2 = -1 ± 2i.
Solving methods without formula
Factorization (decomposition): x²-5x+6 = (x-2)(x-3). Search a,b: a+b=-5, a×b=6 → -2,-3. Completing the square: x²+6x+5 = (x+3)²-9+5 = (x+3)²-4 = 0 → x+3=±2 → x=−1 or x=-5. Mnemonic: "minus b plus minus delta divided by two a."
FAQ
When does a quadratic equation have no solution?
When Δ < 0: no real roots. Example: x²+1=0 (Δ=-4). There is no real number x for which x² = -1. Complex roots: x = ±i (imaginary unit i = √(-1)). Geometrically: the parabola y=x²+1 does not intersect the OX axis.
What are complex elements?
i = √(-1) (imaginary unit). Complex number: a + bi (a = real, b = imaginary). Δ < 0: x = (-b ± i√|Δ|) / (2a). These are pairwise conjugate numbers: x1 = p+qi, x2 = p-qi. Applications: signal theory, quantum mechanics, alternating current. At school: usually only real roots.
How to solve a quadratic equation by factorization?
Patterns: x²-a² = (x-a)(x+a). x²+2ax+a² = (x+a)². x²-2ax+a² = (x-a)². Method: Can you find p,q such that p+q=b and p×q=c (for a=1)? x²-7x+12 = (x-3)(x-4) (3+4=7, 3×4=12). Not always possible with whole numbers.
What is a parabola and how does it relate to Δ?
y = ax²+bx+c: a parabola in the coordinate system. Δ > 0: the parabola intersects OX at 2 points = 2 roots. Δ = 0: the parabola touches OX = 1 double root. Δ < 0: the parabola does not intersect OX = 0 real roots. Arms: up (a>0), down (a<0). Vertex: (-b/2a, -Δ/4a).
How are quadratic equations used in physics?
Vertical projection: h(t) = h0 + v0×t - (g/2)×t². When the object touches the ground: h(t)=0 → quadratic equation in t. Optics: lens v = f/(f-u) → quadratic equation. Electrics: power P = I²×R (I² = square). Ballistics, aerodynamics, mechanics: wherever "the square of the speed" or "the square of the distance" appears.
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