Question

(x+7)/(x-2)+5/4=36/(x2-4)

Answer 1

I hope this helps you

Answer 2

To solve the equation $\frac{x+7}{x-2}+\frac{5}{4}=\frac{36}{x^2-4}$, we'll first find a common denominator and then simplify the equation. The common denominator is $(x-2)(x+2)$ since $(x^2-4)$ can be factored as $(x-2)(x+2)$.

Let's proceed with the steps:

Step 1: Find the common denominator. $\text{Common denominator} = (x-2)(x+2)$

Step 2: Rewrite each fraction with the common denominator.

$\frac{(x+7)(x+2)}{(x-2)(x+2)} + \frac{5(x-2)}{4(x-2)(x+2)} = \frac{36}{(x-2)(x+2)}$

Step 3: Combine the fractions on the left side of the equation. $\frac{(x+7)(x+2) + 5(x-2)}{4(x-2)(x+2)} = \frac{36}{(x-2)(x+2)}$

Step 4: Expand and simplify the numerator on the left side.

$(x+7)(x+2) + 5(x-2) = x^2 + 2x + 7x + 14 + 5x - 10$ $= x^2 + 14x + 4$

Now the equation becomes: $\frac{x^2 + 14x + 4}{4(x-2)(x+2)} = \frac{36}{(x-2)(x+2)}$

Step 5: Cross-multiply to eliminate the denominators.

$36(x^2 + 14x + 4) = 4(x-2)(x+2)$

Step 6: Expand and simplify.

$36x^2 + 504x + 144 = 4(x^2 - 4)$

Step 7: Expand the right side.

$36x^2 + 504x + 144 = 4x^2 - 16$

Step 8: Move all terms to one side of the equation to set it to zero.

$36x^2 + 504x + 144 - 4x^2 + 16 = 0$

Step 9: Combine like terms.

$32x^2 + 504x + 160 = 0$

Step 10: Divide the whole equation by 16 to simplify it further.

$2x^2 + 31.5x + 10 = 0$

Step 11: Solve the quadratic equation using factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

where $a = 2$, $b = 31.5$, and $c = 10$.

$x = \frac{-31.5 \pm \sqrt{31.5^2-4 \cdot 2 \cdot 10}}{2 \cdot 2}$

Now, calculate the value of $x$:

$x = \frac{-31.5 \pm \sqrt{992.25 - 80}}{4}$

$x = \frac{-31.5 \pm \sqrt{912.25}}{4}$

$x = \frac{-31.5 \pm 30.2}{4}$