Question

find x from the following equation using properties of proportion: x2-x+1 \x2 +x+1 = 14 (x-1)\13(x+1)
x2 means x square

Answer 1

IF a/b= c/d
then using property a+b/b=c+d/d
then cross multiply we get x=3

Answer 2

The value of x is

x = 3

• Given equation,

$\frac{ {x}^{2} - x + 1}{ {x}^{2} + x + 1} = \frac{14(x - 1)}{13(x + 1)}$

$\frac{ {x}^{2} - x + 1}{ {x}^{2} + x + 1} = \frac{14x - 14}{13x + 13} \: \: \: - (1)$

• Now, using the componendo dividendo property that is

$if \: \frac{a}{b} = \frac{c}{d} \\ then \: \frac{a + b}{a - b} = \frac{c + d}{c - d}$

• We can write (1) as

$\frac{ ({x}^{2} - x + 1) + ( {x}^{2} + x + 1) }{( {x}^{2} - x + 1) - ( {x}^{2} + x + 1)} = \frac{(14x - 14) + (13x + 13)}{(14x - 14) - (13x + 13)}$

$\frac{ {x}^{2} - x + 1 + {x}^{2} + x + 1}{{x}^{2} - x + 1- {x}^{2} - x - 1} = \frac{14x - 14 + 13x + 13}{14x - 14 - 13x - 13}$

$\frac{2 {x}^{2} + 2}{ - 2x} = \frac{27x -1}{x - 27}$

$(x - 27)( {x}^{2} + 1) = ( - x)(27x - 1)$

${x}^{3} - 27 {x}^{2} + x - 27 = - 27 {x}^{2} + x$

${x}^{3} - 27 {x}^{2} + 27 {x}^{2} + x - x = 27$

${x}^{3} = 27$

therefore,

$x = 3$