Question

solve withoit spamming​

Answer 1

Answer:

$1) \frac{1}{a + b + x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x} \\ \frac{1}{ a+ b + x} = \frac{bx + ax + ab}{abx} \\ cross \: multiply \\ abx = (a + b+ c)( bx+ ax +ab ) \\ abx =a ( bx+ ax +ab ) +b (bx + ax + ab) + x(bx + ax +ab ) \\ abx = abx + {a}^{2} x + {a}^{2} b + {b}^{2} x + abx + a {b}^{2} + b {x}^{2} + a {x}^{2} + abx \\ abx - abx = (abx + {a}^{2} x )+ ({a}^{2} b + a{b}^{2} ) + ({b}^{2} x + abx )+ (b{x}^{2} + a {x}^{2} ) \\ 0 = ax( b+ a) +ab ( a+ b) +bx ( b+ a) + {x}^{2} (b + a) \\ 0 = (b + a)( ax+ ab +bx + {x}^{2} ) \\ Since \: a + b ≠0 \\Therefore, \: ax + ab+ bx + {x}^{2} = 0 \\ ( {x}^{2} + ax) +( bx + ab) = 0 \\ x(x + a) + b(x + a) = 0 \\ (x +a )(x + b) = 0 \\ (x + a = 0)(x + b = 0) \\ x = ( - a) \: and \: x = ( - b)$

$2) \frac{1}{x - 2} + \frac{2}{x - 1} = \frac{6}{x} \\ \frac{1(x - 1) + 2(x - 2)}{(x - 2)(x - 1)} = \frac{6}{x} \\ \frac{x - 1 + 2x - 4}{x(x - 1) - 2(x - 1)} = \frac{6}{x} \\ \frac{x + 2x - 1 - 4}{ {x}^{2} - x - 2x + 2 } = \frac{6}{x} \\ \frac{3x - 5}{ {x}^{2} - 3x + 2 } = \frac{6}{x} \\ cross \: multiply \\ 6( {x}^{2} - 3x + 2) = x(3x - 5) \\ 6 {x}^{2} - 18x + 12 = 3 {x}^{2} - 5x \\ 6 {x}^{2} - 3 {x}^{2} - 18x + 5x + 12 = 0 \\ 3 {x}^{2} - 13x + 12 = 0 \\ 3{x}^{2} - 9x - 4x + 12 = 0 \\ (3 {x}^{2} - 9x) -( 4x + 12) = 0 \\ 3x(x - 3) - 4(x - 3) = 0 \\ (x - 3)(3x - 4) = 0 \\ (x - 3 = 0)(3x - 4 = 0) \\ x = 3 \: and \:x = \frac{4}{3} \\ Therefore, \: x = 3$