Question

用配方法解方程：（1）x平方+x-1=0 (2)2/3x 平方+1/3x=2

Answer 1

(1) x平方+x -1=0
x² + x = 1
x² + x + 1/4 = 1 + 1/4
(x + 1/2)² = 5/4
x = -1/2 ± √5/2 = [-1 ± √5] / 2

(2) 2/3x 平方+1/3x=2
2x²/3 + x/3 = 2
2x² + x = 6
x² + x/2 = 3
(x + 1/4)² = 3 + 1/16 = (7/4)²
x = -1/4 ± √5/2 = [-1 ± 7] / 4
x = 3/2 或者 x = - 2

一般配方法的本质： Ax² + Bx + C = 0
A[x² + Bx/A + (B/2A)²] + C - A*(B/2A)² = 0
A[x + B/2A]²= B²/4A - C = [B² - 4AC] / (4A)
[x + B/2A]² = [B² - 4AC] / (4A²)
x + B/2A = ±√B² - 4AC) / (2A)
则 (x - x1) (x - x2) = 0
其中： x12 = [- B ±√B² - 4AC)] / (2A)