Question

Chứng minh các pt sau luôn có nghiệm :

a, \(x^2-\left(m+1\right)x+m=0\)

b, \(x^2-2\left(m+1\right)+2m+1=0\)

c, \(x^2+\left(m+3\right)x+m+1\) = 0

d, \(x^2-3x+1-m^2=0\)

Answer 1

a,\(x^2-\left(m+1\right)x+m=0\)

xét \(\Delta=\left\{-\left(m+1\right)\right\}^2-4\cdot1\cdot m=m^2+2m+1-4m=m^2-2m+1=\left(m-1\right)^2\ge0\forall m\)

vậy ...

b,\(x^2-2\left(m+1\right)x+2m+1=0\)

xét \(\Delta=\left\{-2\left(m+1\right)\right\}^2-4\cdot1\cdot\left(2m+1\right)=4m^2+8m+4-8m-4=4m^2\ge0\forall m\)

vậy ...

c, \(x^2+\left(m+3\right)x+m+1=0\)

xét \(\Delta=\left(m+3\right)^2-4\cdot1\cdot\left(m+1\right)=m^2+6m+9-4m-4=m^2-2m+5=m^2-2m+1+4=\left(m-1\right)^2+4>0\forall m\)vậy ...

d,\(x^2+3x+1-m^2=0\)

xét \(\Delta=3^2-4\cdot1\cdot\left(1-m^2\right)=9-4+4m^2=4m^2+5>0\forall m\)vậy ...