6 divided by the difference of a number and 2, minus 5 divided by a number plus 2, equals 5 times the reciprocal of the difference of the number squared and 4. What is the number? Please help ASAP!!!!!!!!! :(
Well, it looks like that equation is trying to make your brain do some gymnastics! But don't worry, I'm here to help you out with a smile on my face.
Let's break down the equation step by step and make it less complicated, shall we?
6 / (x - 2) - 5 / (x + 2) = 5 * (1 / (x^2 - 4))
Hmm, when I see fractions and division, my clown instincts kick in! Let's find a common denominator to make things simpler.
The common denominator for the fractions on the left side of the equation is (x - 2)(x + 2). So, we multiply the numerators and denominators accordingly.
[6(x + 2) - 5(x - 2)] / [(x - 2)(x + 2)] = 5 / (x^2 - 4)
Now, let's simplify further and distribute the terms:
[6x + 12 - 5x + 10] / [(x - 2)(x + 2)] = 5 / (x^2 - 4)
Combining like terms:
[x + 22] / [(x - 2)(x + 2)] = 5 / (x^2 - 4)
To get rid of the denominator on both sides, let's cross-multiply:
5(x - 2)(x + 2) = (x + 22)(x^2 - 4)
Still with me? Great! Now let's expand and simplify that equation:
5(x^2 - 4) = (x + 22)(x^2 - 4)
5x^2 - 20 = x^3 + 18x^2 - 4x^2 - 88
Combine like terms again:
5x^2 - 20 = x^3 + 14x^2 - 88
Let's bring everything to one side:
x^3 + 9x^2 - 68 = 0
Phew! Now we have a cubic equation. Unfortunately, solving it without further restrictions or information would be a bit tricky to do in a jiffy.
So, my recommendation is to take a deep breath, grab a pencil and paper, and use numerical or algebraic methods to solve for x. If you need assistance in solving cubic equations, there are plenty of handy videos and online resources available.
Remember, math can be a clown's worst nightmare, but with some patience and perseverance, you can conquer it! Good luck!
Let's break down the problem step-by-step.
Step 1: Assign a variable to the unknown number.
Let's assume the unknown number is represented by the variable x.
Step 2: Translate the given information into an equation.
According to the problem, we have the following equation:
6 / (x - 2) - 5 / (x + 2) = 5 * (1 / (x^2 - 4))
Step 3: Simplify the equation.
To simplify the equation, we need to get rid of the fractions by finding a common denominator. In this case, the common denominator is (x - 2)(x + 2)(x^2 - 4).
Multiplying every term by the common denominator, the equation becomes:
6(x + 2)(x^2 - 4) - 5(x - 2)(x^2 - 4) = 5(x - 2)(x + 2)
Step 4: Expand and simplify the equation.
Now, let's expand and simplify the equation:
6(x^3 + 2x^2 - 4x - 8) - 5(x^3 - 2x^2 - 4x + 8) = 5(x^2 - 4)
6x^3 + 12x^2 - 24x - 48 - 5x^3 + 10x^2 + 20x - 40 = 5x^2 - 20
x^3 + 2x^2 - 4x - 8 = 0
Step 5: Solve the equation.
To find the value of x, we need to solve the equation x^3 + 2x^2 - 4x - 8 = 0.
Unfortunately, there is no simple way to solve this equation algebraically. You can either use numerical methods like graphing or approximations to find the solutions.
To solve this equation, let's break it down step by step:
Step 1: Let's assign a variable to represent the unknown number. Let's call it "x".
Step 2: Translate the given information into an equation:
The equation is: 6 / (x - 2) - 5 / (x + 2) = 5 * (1 / (x^2 - 4))
Step 3: Simplify the equation:
The first step is to find the least common denominator (LCD) of the fractions on the left side of the equation. In this case, it is (x - 2)(x + 2):
(6(x + 2) - 5(x - 2)) / ((x - 2)(x + 2)) = 5 * (1 / (x^2 - 4))
Simplifying further:
(6x + 12 - 5x + 10) / ((x - 2)(x + 2)) = 5 / (x^2 - 4)
(x + 22) / ((x - 2)(x + 2)) = 5 / (x^2 - 4)
Step 4: Eliminate the denominators by cross-multiplying:
(x + 22) * (x^2 - 4) = 5 * ((x - 2)(x + 2))
Simplifying further:
(x^3 - 4x + 22x - 88) = 5 * (x^2 - 4)
Step 5: Expand and simplify the equation:
x^3 + 18x - 88 = 5x^2 - 20
Rearranging the equation:
x^3 - 5x^2 + 18x + 20 - 88 = 0
x^3 - 5x^2 + 18x - 68 = 0
Step 6: Solve the equation:
Unfortunately, there isn't a simple, straightforward way to find the solutions to a cubic equation. You can use numerical methods, such as graphing or approximation techniques, to find an approximate solution.
Alternatively, you can use online equation solvers or specialized math software to find the exact solutions.
I recommend using a graphing calculator or an online equation solver to find the values of "x."
I hope this helps!
[6 / (n - 2)] - [5 / (n + 2)] = 5 / (n^2 - 4)
... (n^2 - 4) is the LCD
(6 n + 12) - (5 n - 10) = 5