Respuesta :
We start with
[tex]\dfrac{x}{x+4} = -\dfrac{9}{x+3}[/tex]
Assuming that [tex]x \notin \{-4,\ -3\}[/tex] we can cross multiply the equation:
[tex]x(x+3) = -9(x+4) \iff x^2+3x=-9x-36 \iff x^2+12x+36=0[/tex]
You can recognize the perfect square pattern in the quadratic equation:
[tex]x^2+12x+36 = (x)^2 + 2\cdot x \cdot 6 + (6)^2 = (x+6)^2[/tex]
So, we have
[tex]x^2+12x+36=0 \iff (x+6)^2 = 0 \iff x+6=0 \iff x=-6[/tex]
Which is the only solution, with multiplicity 2.
The value of x is -6 and this can be determined by simplifying the given expression by using the arithmetic operations.
Given :
Expression -- [tex]\dfrac{x}{x+4}=-\dfrac{9}{x+3}[/tex]
The following steps can be used in order to determine the values of 'x':
Step 1 - The arithmetic operations can be used in order to determine the values of 'x'.
Step 2 - Write the given expression.
[tex]\dfrac{x}{x+4}=-\dfrac{9}{x+3}[/tex]
Step 3 - Cross multiply in the above expression.
[tex]x(x+3)=-9(x+4)[/tex]
Step 4 - Further simplify the above expression.
[tex]x^2+3x+9x+36=0[/tex]
[tex]x^2+12x+36=0[/tex]
Step 5 - Factorize the above quadratic equation.
[tex]x^2+6x+6x+36=0[/tex]
x(x + 6) + 6(x + 6) = 0
[tex](x+6)^2=0[/tex]
So, the value of 'x' is -6.
For more information, refer to the link given below:
https://brainly.com/question/17177510
