Question

3(3+X)+6(1+x)/2x=3/2​

Answer 1

Answer:

$\boxed{x = \frac{ - 7 \pm \sqrt{33} }{4}}$

Step-by-step explanation:

$Solve \: for \: x \: over \: the \: real \: numbers: \\ 3(3 + x) + \frac{6(x + 1)}{2x} = \frac{3}{2} \\ \\ \frac{6(x + 1)}{2x} = \frac{3(x + 1)}{x}: \\ = > 3(3 + x) + \boxed{\frac{3(x + 1)}{x}} = \frac{3}{2} \\ \\ Bring \: \: 3(3 + x) + \frac{3(x + 1)}{x} \:together \\ using \: the \: common \: denominator \: x: \\ = > (3(3 + x) \times \frac{x}{x} ) + \frac{3(x + 1)}{x} = \frac{3}{2} \\ = > \frac{9x + 3 {x}^{2} }{x} + \frac{3(x + 1)}{x} = \frac{3}{2}\\ = > \frac{9x + 3 {x}^{2} }{x} + \frac{3x + 3}{x} = \frac{3}{2}\\ = > \frac{(9x +3 {x}^{2} ) + (3x + 3)}{x} = \frac{3}{2}\\ = > \frac{3 {x}^{2} + 9x + 3x + 3 }{x} = \frac{3}{2} \\ = > \frac{3 {x}^{2} + 12 + 3 }{x} = \frac{3}{2}\\ = > \frac{3( {x}^{2} + 4x + 1 )}{x} = \frac{3}{2}$

$\\ Cross \: multiply: \\ = > (3( {x}^{2} + 4x + 1)) \times 2 = 3 \times x \\ = > 6( {x}^{2} + 4x + 1) = 3x \\ \\ Expand \: out \: terms \: of \: the \: left \: hand \: side: \\ = > 6 {x }^{2} + 24x + 6 = 3x \\ \\ Subtract \: 3 x \: from \: both \: sides: \\ = > 6 {x}^{2} + 21x + 6 = 0 \\ \\ Divide \: both \: sides \: by \: 6: \\ = > {x}^{2} + \frac{7x}{2} + 1 = 0 \\ \\ Subtract \: 1 \: from \: both \: sides: \\ = > {x}^{2} + \frac{7x}{2} = - 1 \\ \\ Add \: \frac{49}{16} \: to \: both \: sides: \\ = > {x}^{2} + \frac{7x}{2} + \frac{49}{16} = \frac{33}{16} \\ \\ Write \: the \: left \: hand \: side \: as \: a \: square: \\ = > { (x + \frac{7}{4} )}^{2} = \frac{33}{16} \\ \\ Take \: the \: square \: root \: of \: both \: sides: \\ = > x + \frac{7}{4} = \pm \frac{ \sqrt{33} }{4} \\ \\ Subtract \: \frac{7}{4} \: from \: both \: sides: \\ = > x = - \frac{7}{4} \pm \frac{ \sqrt{33} }{4} \\ = > x = \frac{ - 7 \pm \sqrt{33} }{4}$