Question

Identify all values of x that make the equation true.a. 2x+1/x=1/x-2b. 1/x+2=2/x-1c. x+3/1-x=x+1/x+2d. x+2/x+8=1/x+2

Answer 1

Given:

a)

`\begin{gathered} (2x+1)/(x)=(1)/(x-2) \\ (2x+1)(x-2)=x*1 \\ 2x^2-4x+x-2=x \\ 2x^2-4x-2=0 \\ \text{Use quadratic formula: ax}^2+bx+c=0 \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x=(-\left(-4\right)\pm√(\left(-4\right)^2-4\cdot\:2\left(-2\right)))/(2\cdot\:2) \\ x=(-\left(-4\right)\pm\:4√(2))/(2\cdot\:2) \\ x=1+√(2),\: x=1-√(2) \end{gathered}`

b)

`\begin{gathered} (1)/(x+2)=(2)/(x-1) \\ x-1=2(x+2) \\ x-1=2x+4 \\ x=-5 \end{gathered}`

c)

`\begin{gathered} (x+3)/(1-x)=(x+1)/(x+2) \\ (x+3)(x+2)=(x+1)(1-x) \\ x^2+2x+3x+6=1-x^2 \\ 2x^2+5x+5=0 \\ \text{Use quadratic formula,} \\ x=(-5\pm√(5^2-4\cdot\:2\cdot\:5))/(2\cdot\:2) \\ x=(-5\pm√(15)i)/(2\cdot\:2) \\ x=(-5+√(15)i)/(2\cdot\:2),\: x=(-5-√(15)i)/(2\cdot\:2) \\ x=-(5)/(4)+i(√(15))/(4),\: x=-(5)/(4)-i(√(15))/(4) \end{gathered}`

d)

`\begin{gathered} (x+2)/(x+8)=(1)/(x+2) \\ (x+2)(x+2)=(x+8) \\ x^2+4x+4=x+8 \\ x^2+3x-4=0 \\ x^2-x+4x-4=0 \\ x(x-1)+4(x-1)=0 \\ (x-1)(x+4)=0 \\ x-1=0,x+4=0 \\ x=1,-4 \end{gathered}`