How do you solve #x+4=-4/x#?

Question

Savannah Anderson · Accepted Answer

The solution is #color(blue)(x=-2#

#x+4=-4/color(blue)(x#

#color(blue)(x) * (x+4)=-4#

#x^2 +4x =-4#

#x^2 +4x +4=0#

We can Split the Middle Term of this expression to factorise it and thereby find the solution.

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 = 1*4 =4# AND #N_1 +N_2 = b =4#

After trying out a few numbers we get #N_1 = 2# and #N_2 =2# #2*2= 4#, and #2+2=4#

#x^2 +color(blue)(4x) +4=x^2 +color(blue)(2x+2x) +4#

#x(x+2) +2(x +2)=0#

#(x+2)(x+2) =0#

We now equate the factor to zero to obtain the solution(both factors are equal here):

#x+2=0, color(blue)(x=-2#

Isaiah Anthony · Answer

undefined To solve the equation x + 4 = -4/x, we can start by multiplying both sides of the equation by x to eliminate the fraction. This gives us x(x + 4) = -4. Expanding the left side of the equation, we get x^2 + 4x = -4. Rearranging the equation, we have x^2 + 4x + 4 = 0. Factoring this quadratic equation, we find (x + 2)^2 = 0. Taking the square root of both sides, we get x + 2 = 0. Solving for x, we find x = -2. Therefore, the solution to the equation x + 4 = -4/x is x = -2.