Question

X^2+16/x^2=256,найти x+4/x

Answer 1

Решение задания приложено

Answer 2

To find the value of $x + \frac{4}{x}$ when $x^2 + \frac{16}{x^2} = 256$, we can follow these steps:

Step 1: Let's simplify the given equation. To do this, first, multiply the entire equation by $x^2$ to eliminate the fraction:

$x^2 \cdot \left(x^2 + \frac{16}{x^2}\right) = x^2 \cdot 256$

Step 2: Expand the equation:

$x^4 + 16 = 256x^2$

Step 3: Move all terms to one side of the equation:

$x^4 - 256x^2 + 16 = 0$

Step 4: This is a quadratic equation in terms of $x^2$. Let's solve for $x^2$ using the quadratic formula:

The quadratic formula states that for an equation of the form $ax^2 + bx + c = 0$, the solutions for $x$ are given by:

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

In our equation, $a = 1$, $b = -256$, and $c = 16$:

$x^2 = \frac{256 \pm \sqrt{(-256)^2 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

$x^2 = \frac{256 \pm \sqrt{65536 - 64}}{2}$

$x^2 = \frac{256 \pm \sqrt{65472}}{2}$

Step 5: Now, calculate the two possible values of $x^2$:

$x^2 = \frac{256 + \sqrt{65472}}{2}$

$x^2 = \frac{256 - \sqrt{65472}}{2}$

Step 6: Calculate the square roots:

$x = \sqrt{\frac{256 + \sqrt{65472}}{2}}$

$x = \sqrt{\frac{256 - \sqrt{65472}}{2}}$

Step 7: Finally, we can find $x + \frac{4}{x}$ for each solution:

For $x = \sqrt{\frac{256 + \sqrt{65472}}{2}}$:

$x + \frac{4}{x} = \sqrt{\frac{256 + \sqrt{65472}}{2}} + \frac{4}{\sqrt{\frac{256 + \sqrt{65472}}{2}}}$