Solve the following equation for the variable x.
(4/x-2)+(3/x+2)=(-3x/x^2-4)
Thank you very much for your help.
To solve the equation:
(4/x-2) + (3/x+2) = (-3x/x^2-4).
First, let's find a common denominator for the fractions:
For the first fraction, (4/x-2), the denominator is (x-2).
For the second fraction, (3/x+2), the denominator is (x+2).
For the third fraction, (-3x/x^2-4), the denominator is (x^2-4).
Since (x^2-4) can be factored as (x+2)(x-2), the least common denominator is (x+2)(x-2).
Now, let's rewrite the equation using the common denominator:
(4(x+2) + 3(x-2))/(x+2)(x-2) = (-3x)/(x+2)(x-2).
Next, let's simplify the numerators:
(4x+8 + 3x-6)/(x+2)(x-2) = (-3x)/(x+2)(x-2).
Combining like terms:
(7x+2)/(x+2)(x-2) = (-3x)/(x+2)(x-2).
Now, let's remove the common factors (x+2)(x-2) from the equation:
7x+2 = -3x.
Let's gather all the x terms on one side:
7x + 3x = -2.
Combining like terms:
10x = -2.
Finally, let's solve for x by dividing both sides by 10:
x = -2/10.
Simplifying the fraction gives us the final answer:
x = -1/5.
To solve the given equation (4/x-2)+(3/x+2)=(-3x/x^2-4) for the variable x, follow these steps:
Step 1: Remove the denominators.
To clear the denominators, we need to multiply every term in the equation by the least common denominator (LCD). The LCD of the given equation is (x - 2)(x + 2)(x^2 - 4).
Multiplying each term by the LCD gives us:
(x - 2)(x + 2)(x^2 - 4) * [(4/x-2)+(3/x+2)]= (-3x/x^2-4) * (x - 2)(x + 2)(x^2 - 4)
Simplifying this equation will help to eliminate the denominators.
Step 2: Simplify the equation.
Multiply the LCD with each term in the equation:
(x - 2)(x + 2)(x^2 - 4) * (4/x - 2) + (x - 2)(x + 2)(x^2 - 4) * (3/x + 2) = (-3x/x^2 - 4) * (x - 2)(x + 2)(x^2 - 4)
Now, distribute and simplify both sides of the equation. This involves canceling out common factors and rearranging terms:
(x^2 - 4) * 4 + (x^2 - 4) * 3 = -3x * (x^2 - 4)
4x^2 - 16 + 3x^2 - 12 = -3x^3 + 12x
Combine like terms:
7x^2 - 28 = -3x^3 + 12x
Step 3: Arrange the equation in standard form.
Rearrange the equation to have the highest exponent term on one side, and the remaining terms on the other side:
3x^3 + 7x^2 - 12x - 28 = 0
Now, the equation is in standard form.
Step 4: Solve the equation.
To solve the equation, you can use numerical or algebraic methods. One way to find the solutions is by factoring, if possible, or by using the Newton-Raphson method, the bisection method, or other numerical methods available in calculators or computer software.
Note that factoring a cubic equation can sometimes be challenging, and in such cases, numerical methods are often preferred.
With the solutions found, the values of x will satisfy the given equation.
Please note that the solution to this specific equation requires further mathematical calculations that may be lengthy and complex. It is recommended to use appropriate mathematical software or calculators to get accurate numerical solutions.