Question

用配方法解ax的二次方加bx加c等于零

Answer 1

◆用配方法解ax²+bx+c=0. (a≠0)
解:原方程两边同除以a,得:x²+(b/a)x+c/a=0.
配方,得:[x²+(b/a)x+(b/2a)²]-(b/2a)²+c/a=0.
即:(x+b/2a)²=b²/4a²-c/a=(b²-4ac)/4a².
当b²-4ac<0时,二次方程无解;
当b²-4ac=0时,(x+b/2a)²=0, x1=x2=-b/2a;
当b²-4ac>0时,(x+b/2a)²=(b²-4ac)/4a²,x+b/2a=±√(b²-4ac)/2a.
即:x1=-b/2a+√(b²-4ac)/2a=[-b+√(b²-4ac)]/2a;
x2=-b/2a-√(b²-4ac)/2a=[-b-√(b²-4ac)]/2a.

Answer 2

ax^2+bx+c=0
a(x^2+bx/a)+c=0
a[x^2+bx/a+b^2/(4a^2)-b^2/(4a^2)]+c=0
a(x^2+bx/a+b^2/4a)-b^2/4a+c=0
a[x+b/(2a)]^2+(4ac-b^2)/4ac=0
解得x=(-b加减根号下△)/2a