Kalkulator uzupełniania do kwadratu (postać kanoniczna)
Ten kalkulator zapisuje ax² + bx + c w postaci kanonicznej a(x − h)² + k. Podaje wierzchołek paraboli i dokładne rozwiązania ax² + bx + c = 0, pokazując każdy krok metody.
Wynik
Wpisz wartości i naciśnij Oblicz, aby zobaczyć wynik.
Wzór
Halve the x coefficient, square it, add and subtract it. The vertex is (−b/2a, c − b²/4a).
Jak korzystać z tego kalkulatora
- Write the quadratic as ax² + bx + c and enter a, b and c (a cannot be 0).
- Press Calculate to see the vertex form, the vertex and the solutions.
Rozwiązane przykłady
Complete the square for x² + 6x + 5
Half of 6 is 3, and 3² = 9.
x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4.
The vertex is (−3, −4). Solving (x + 3)² = 4 gives x = −3 ± 2, so x = −5 or x = −1.
Irrational solutions: x² − 4x + 1
Half of −4 is −2, and (−2)² = 4.
x² − 4x + 1 = (x − 2)² − 3, so x = 2 ± √3.
With a ≠ 1: 2x² − 12x + 7
Factor out 2: 2(x² − 6x) + 7.
2[(x − 3)² − 9] + 7 = 2(x − 3)² − 11. The vertex is (3, −11).
(x − 3)² = 11/2, so x = 3 ± √(11/2) = (6 ± √22)/2.
No real solutions: x² + 2x + 5
x² + 2x + 5 = (x + 1)² + 4.
A square is never negative, so (x + 1)² + 4 = 0 has no real solution.
Częste błędy
- Forgetting to subtract the square you added, which changes the expression.
- Halving b without first dividing by a when a ≠ 1.
- Sign of the vertex: (x + 3)² − 4 has its vertex at x = −3, not 3.
Najczęstsze pytania
How do you complete the square?
Make the x² coefficient 1, take half of the x coefficient and square it, then add and subtract that number. x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4.
How do you find the vertex by completing the square?
Write the quadratic as a(x − h)² + k. The vertex (turning point) is (h, k): for (x + 3)² − 4 it is (−3, −4).
Is completing the square the same as the quadratic formula?
The quadratic formula comes from completing the square on ax² + bx + c = 0, so both give the same solutions.