Question

(AM GIVING BRAINLIEST) Find all values of x such that x/(x - 5) =4/(x - 4)

Answer 1

Answer:

x = imaginary number

x₁ = 2i

x₂ = 6i

Step-by-step explanation:

x/(x - 5) =4/(x - 4)

x(x-4) = 4(x-5)

x*x + x*-4 = 4*x * 4*-5

x² -4x = 4x - 20

x² -4x - 4x + 20 = 0

x² - 8x + 20 = 0

x = {-(-8)±√((-8²)-(4*1*20))} / (2*1)

x = {8±√(64-80)} / 2

x = {8±√-16} / 2

√(-16) ∈ i

i =  set of imaginary numbers

√-16 = -4i

then:

x = {8±(-4i) } / 2

x₁ = {8+(-4i)}/2 = {8-4i)/2 = 4i/2 = 2i

x₂ = {8-(-4)}/2 = {8+4i}/2 = 12i/2 = 6i

Answer 2

Answer:

$\boxed{\sf x = 4 \pm 2i}$

Step-by-step explanation:

$\sf Solve \: for \: x: \\ \sf \implies \frac{x}{x - 5} = \frac{4}{x - 4} \\ \\ \sf Cross \: multiply: \\ \sf \implies x(x - 4) = 4(x - 5) \\ \\ \sf Expand \: out \: terms \: of \: the \: left \: hand \: side: \\ \sf \implies {x}^{2} - 4x = 4(x - 5) \\ \\ \sf Expand \: out \: terms \: of \: the \: right \: hand \: side: \\ \sf \implies {x}^{2} - 4x = 4x - 20 \\ \\ \sf Subtract \: 4 x \: from \: both \: sides:\\ \sf \implies {x}^{2} - 4x - 4x = 4x - 20 - 4x \\ \\ \sf \implies {x}^{2} - 8x = - 20\\ \\ \sf Add \: 16 \: to \: both \: sides: \\ \sf \implies {x}^{2} - 8x + 16 = - 20 + 16 \\ \\ \sf\implies {x}^{2} - 8x + 16 = -4 \\ \\ \sf Write \: the \: left \: hand \: side \: as \: a \: square: \\ \sf\implies {x}^{2} - 4x - 4x + 16 = -4 \\ \\ \sf\implies x(x - 4) - 4(x - 4) = -4 \\ \\ \sf\implies (x - 4)(x - 4) = -4 \\ \\ \sf\implies {(x - 4)}^{2} = - 4 \\ \\ \sf Take \: the \: square \: root \: of \: both \: sides: \\ \sf\implies \sqrt{ {(x - 4)}^{2}} = \sqrt{ - 4} \\ \\ \sf\implies x - 4 = \pm 2i \\ \\ \sf Add \: 4 \: to \: both \: sides: \\ \sf\implies x - 4 + 4= \pm 2i + 4 \\ \\ \sf\implies x = 4\pm 2i$