요금제

완전제곱식 만들기 계산기 (꼭짓점 꼴)

이 계산기는 ax² + bx + c를 꼭짓점 꼴 a(x − h)² + k로 바꿔 씁니다. 포물선의 꼭짓점과 ax² + bx + c = 0의 정확한 해를 구하며, 방법의 각 단계를 보여 줍니다.

결과

값을 입력하고 계산을 누르면 결과가 표시됩니다.

공식

ax2+bx+c=a(x+b2a)2+c−b24aax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}

Halve the x coefficient, square it, add and subtract it. The vertex is (−b/2a, c − b²/4a).

이 계산기 사용법

  1. Write the quadratic as ax² + bx + c and enter a, b and c (a cannot be 0).
  2. Press Calculate to see the vertex form, the vertex and the solutions.

풀이 예제

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.

자주 하는 실수

  • 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.

자주 묻는 질문

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.