Question

5^((x^2)-(5/7)x)=корень 7 степени из 25

Answer 1

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Answer 2

To solve the equation $5^{x^2 - \frac{5}{7}x} = \sqrt[7]{25}$, we'll start by simplifying the right-hand side:

$\sqrt[7]{25} = 25^{1/7}$

Next, we'll rewrite 25 as a power of 5 because the equation involves $5^x$:

$25 = 5^2$

Therefore:

$\sqrt[7]{25} = 5^{2/7}$

Now the equation becomes:

$5^{x^2 - \frac{5}{7}x} = 5^{2/7}$

Since the bases are the same (both are 5), the exponents must be equal:

$x^2 - \frac{5}{7}x = \frac{2}{7}$

Now, we'll solve this quadratic equation for $x$:

$x^2 - \frac{5}{7}x - \frac{2}{7} = 0$

To make it easier to factor, we can multiply the entire equation by 7:

$7x^2 - 5x - 2 = 0$

Now, we'll try to factor the quadratic expression:

$(7x + 2)(x - 1) = 0$

Now, set each factor to zero and solve for $x$:

1. $7x + 2 = 0$

$7x = -2$

$x = \frac{-2}{7}$

2. $x - 1 = 0$

$x = 1$

So, the solutions to the equation are $x = \frac{-2}{7}$ and $x = 1$.