How do you multiply # 1 / (x(x - 2))+ x /(x - 2) = 10/ x#?

Question

How do you multiply # 1 / (x(x - 2))+   x /(x - 2) = 10/ x#?

Nathan Axel · Accepted Answer

# 1 / (x(x - 2))+ x /(x - 2) = 10/ x# L.C.M of # x(x-2) , (x-2) and x = x(x-2)#  # 1 / (x(x - 2))+ (x xx x) /((x - 2) xx x) = (10 xx (x-2))/ (x xx (x-2))# # (1 + x^2) /( x(x - 2))  = (10x - 20)/ (x (x-2))# # 1 + x^2  = 10x - 20# # x^2 - 10x +21 = 0# We can Split the Middle Term of this expression to factorise it
In this technique, if we have to factorise an expression like #ax^2 + bx + c#, we need to think of 2 numbers such that: #N_1*N_2 = a*c = 1xx21 = 21# And #N_1 +N_2 = b = -10#
After trying out a few numbers we get : #N_1 = -3# and #N_2 =-7#
#-3 xx-7 = 21#, and #(-3) + (-7)= -10#  # x^2 - 10x +21 =x^2 - 3x - 7x+21  # # = x(x-3) - 7(x-3)#
# = (x-3) (x-7)# the solutions are # x = 3 , x = 7#

Jace Anderson · Answer

undefined To solve the equation 1 / (x(x - 2)) + x / (x - 2) = 10 / x, we can start by finding a common denominator for the fractions on the left side of the equation. The common denominator is x(x - 2).

Multiplying the first fraction by (x - 2) / (x - 2) and the second fraction by x / x, we get:

[(1 * (x - 2)) / (x(x - 2))] + [(x * x) / (x(x - 2))] = 10 / x

Simplifying the fractions, we have:

[(x - 2) / (x^2 - 2x)] + (x^2 / (x^2 - 2x)) = 10 / x

Combining the fractions, we get:

[(x - 2) + x^2] / (x^2 - 2x) = 10 / x

Expanding the numerator, we have:

(x - 2 + x^2) / (x^2 - 2x) = 10 / x

Rearranging the equation, we get:

(x^2 + x - 2) / (x^2 - 2x) = 10 / x

Cross-multiplying, we have:

x(x^2 + x - 2) = 10(x^2 - 2x)

Expanding both sides, we get:

x^3 + x^2 - 2x = 10x^2 - 20x

Rearranging the equation, we have:

x^3 + x^2 - 10x^2 + 20x - 2x = 0

Combining like terms, we get:

x^3 - 9x^2 + 18x - 2x = 0

Simplifying further, we have:

x^3 - 9x^2 + 16x = 0

Factoring out an x, we get:

x(x^2 - 9x + 16) = 0

Factoring the quadratic expression, we have:

x(x - 1)(x - 16) = 0

Therefore, the solutions to the equation are x = 0, x = 1, and x = 16.