Question

1)x^2+20/x^2-4=x-3/x+2-6/2-x 2)4x-3/16x^2-9=0

Answer 1

Ответ:

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Answer 2

Let's solve each equation separately:

1. $\frac{x^2 + 20}{x^2 - 4} = \frac{x - 3}{x + 2} - \frac{6}{2 - x}$

First, we need to find a common denominator for the fractions on the right side of the equation:

$\frac{x^2 + 20}{x^2 - 4} = \frac{(x - 3)(2 - x) - 6(x + 2)}{(x + 2)(2 - x)}$

Now, let's simplify the equation:

$\frac{x^2 + 20}{x^2 - 4} = \frac{-x^2 + 5x - 6 - 6x - 12}{-(x^2 - 4)}$

Now, let's simplify further:

$\frac{x^2 + 20}{x^2 - 4} = \frac{-x^2 - x - 18}{-(x^2 - 4)}$

We can simplify this equation further by multiplying both sides by $-(x^2 - 4)$ to get rid of the denominators:

$(x^2 + 20) = (-x^2 - x - 18) \cdot (x^2 - 4)$

Now, expand the right side of the equation:

$x^2 + 20 = (-x^2 - x - 18) \cdot (x^2 - 4)$

Multiply the polynomials on the right side:

$x^2 + 20 = (-x^4 + 4x^2 - x^3 + 4x^2 - 18x^2 + 72)$

Now, combine like terms:

$x^2 + 20 = (-x^4 - x^3 - 10x^2 + 72)$

Subtract $x^2 + 20$ from both sides:

$0 = -x^4 - x^3 - 10x^2 + 72 - x^2 - 20$

Combine like terms again:

$0 = -x^4 - x^3 - 11x^2 + 52$

Now, this is a quartic equation, and solving it can be more involved. Unfortunately, there is no simple closed-form solution for quartic equations, so you may need to use numerical methods or a computer algebra system to find its roots.

2. $\frac{4x - 3}{16x^2 - 9} = 0$

First, notice that the denominator can be factored as the difference of squares:

$16x^2 - 9 = (4x + 3)(4x - 3)$

Now, set the numerator equal to zero:

$4x - 3 = 0$

Add 3 to both sides:

$4x = 3$

Divide by 4:

$x = \frac{3}{4}$

So, the solution to the equation is $x = \frac{3}{4}$.