Question

(x²- x+1) (x²-x+7) =65​

Answer 1

(x² - x + 1)(x² - x - 7) = 65

Сделаем замену :

x² - x + 1 = m , тогда  x² - x - 7 = m - 8

m(m - 8) = 65

m² - 8m - 65 = 0

D = (- 8)² - 4 * (- 65) = 64 + 260 = 324 = 18²

$m_{1} =\frac{8-18}{2}=-5\\\\m_{2}=\frac{8+18}{2}=13\\\\1)x^{2}-x+1=-5\\\\x^{2}-x+6=0\\\\D=(-1)^{2} -4*6=1-24=-23$

Answer 2

To solve the equation $(x^2 - x + 1)(x^2 - x + 7) = 65$, we can expand the expression and then rearrange the equation to solve for $x$.

Expanding the equation, we have:

$(x^2 - x + 1)(x^2 - x + 7) = 65$

Expanding the left side:

$x^4 - x^3 + 7x^2 - x^3 + x^2 - 7x + x^2 - x + 7 = 65$

Combining like terms:

$x^4 - 2x^3 + 9x^2 - 8x + 7 = 65$

Rearranging the equation:

$x^4 - 2x^3 + 9x^2 - 8x + 7 - 65 = 0$

Simplifying:

$x^4 - 2x^3 + 9x^2 - 8x - 58 = 0$

Now, we can solve this quartic equation using numerical methods or factorization if possible. Unfortunately, the quartic equation does not have an easily factorizable form.

One approach to finding the solutions is to use numerical methods or a graphing calculator. Another option is to use a computer algebra system or specialized software to find the roots.

Please let me know if you would like an approximation of the solutions using numerical methods or if you have any other questions.