Question

(x²-7x+16)(x²-7 x+17)=20​

Answer 1

Ответ:

$x_1=3, x_2 = 4, x_3=\frac{7+\sqrt {5}}{2}, x_4=\frac{7-\sqrt {5}}{2}$

Объяснение:

$(x^2-7x+16)(x^2-7 x+17)=20$

Пусть $t = x^2-7x+16$, тогда

$t(t+1)=20$

$(t+5)(t-4)=0$

1) $t=4$

$x^2-7x+16=4$

$x^2-7x+12=0$

$(x-3)(x-4)=0$

$x_1=3$

$x_2=4$

2) $t=-5$

$x^2-7x+16=-5$

$x^2-7x+21=0$

$D=7^2-21*4=5$

$x_3=\frac{7+\sqrt {5}}{2}$

$x_4=\frac{7-\sqrt {5}}{2}$

Answer 2

To solve the equation $(x^2 - 7x + 16)(x^2 - 7x + 17) = 20$, you can start by multiplying out the expressions on the left side of the equation and then simplify the resulting quadratic equation. Here's how you can do it:

Expand the left side of the equation:

$(x^2 - 7x + 16)(x^2 - 7x + 17) = 20$

Now, multiply the two quadratic expressions:

$x^4 - 7x^3 + 16x^2 - 7x^3 + 49x^2 - 119x + 16x^2 - 119x + 272 = 20$

Combine like terms:

$x^4 - 14x^3 + 81x^2 - 238x + 272 = 20$

Now, subtract 20 from both sides of the equation to set it equal to zero:

$x^4 - 14x^3 + 81x^2 - 238x + 252 = 0$

Now, you have a quartic equation. Unfortunately, quartic equations can be challenging to solve algebraically in general. You may need to use numerical methods or a calculator/computer software to find approximate solutions for this equation.