已知方程m²x²-(3m²-8m)x+2m²-13m+15=0(m为正整数)至少有一个整数根,求m的值.
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![已知方程m²x²-(3m²-8m)x+2m²-13m+15=0(m为正整数)至少有一个整数根,求m的值.](/uploads/image/z/2541354-42-4.jpg?t=%E5%B7%B2%E7%9F%A5%E6%96%B9%E7%A8%8Bm%26%23178%3Bx%26%23178%3B-%283m%26%23178%3B-8m%29x%2B2m%26%23178%3B-13m%2B15%3D0%EF%BC%88m%E4%B8%BA%E6%AD%A3%E6%95%B4%E6%95%B0%EF%BC%89%E8%87%B3%E5%B0%91%E6%9C%89%E4%B8%80%E4%B8%AA%E6%95%B4%E6%95%B0%E6%A0%B9%2C%E6%B1%82m%E7%9A%84%E5%80%BC.)
已知方程m²x²-(3m²-8m)x+2m²-13m+15=0(m为正整数)至少有一个整数根,求m的值.
已知方程m²x²-(3m²-8m)x+2m²-13m+15=0(m为正整数)至少有一个整数根,求m的值.
已知方程m²x²-(3m²-8m)x+2m²-13m+15=0(m为正整数)至少有一个整数根,求m的值.
若m=0,方程无解.
故m不能为0,此时为二次方程
m^2x^2-(3m^2-8m)x+(2m-3)(m-5)=0
十字相乘法:
mx -(2m-3)
mx -(m-5)
故有:[mx-(2m-3)][mx-(m-5)]=0
x1=(2m-3)/m=2-3/m
x2=(m-5)/m=1-5/m
为使x1或x2至少有一个为整数,则m须被3或5整除,则时,为保证x1或x2为正整数,
有m=-1,3,-3,-5.
m²x²-(3m²-8m)x+2m²-13m+15=0
m²x²-(3m²-8m)x+(2m-3)(m-5)=0
(mx-2m+3)(mx-m+5)=0
x=(2m-3)/m或x=(m-5)/m
1.x=(2m-3)/m=2-3/m
因为是整数,所以
m=1,-1,3,-3
2.x=(m-5)/m=1-5/m
m=1,-1,5,-5
所以
m=1,-1,5,-5,3或-3.
△=b²-4ac= (3m²-8m)²-4*m²*(2m²-13m+15)
=9m⁴-48m³+64m²-(8m⁴-52m³+60m²)
=m⁴+4m³+4m²
=m²(m+2)²<...
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△=b²-4ac= (3m²-8m)²-4*m²*(2m²-13m+15)
=9m⁴-48m³+64m²-(8m⁴-52m³+60m²)
=m⁴+4m³+4m²
=m²(m+2)²
x=[(3m²-8m)±√△]/(2m²)
x =(3m²-8m+m²+2m)/(2m²)
=2-3/m
得m=1或者m=3
或
x =(3m²-8m-m²-2m)/(2m²)
=1-5/m
得m=1或者m=5
所以m的值为1、3、5
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