Question

X1+x2=7 x1*x2=2 X1^4 + x2^4=? (^4 степень)

Answer 1

$x_1^4+x_2^4=(x_1^2)^2+(x_2^2)^2=(x_1^2)^2+2x_1^2x_2^2+(x_2^2)^2-2x_1^2x_2^2=\\ \\ =(x_1^2+x_2^2)^2-2(x_1x_2)^2=(x_1^2+2x_1x_2+x_2^2-2x_1x_2)^2-2(x_1x_2)^2=\\ \\ =((x_1+x_2)^2-2x_1x_2)^2-2(x_1x_2)^2=(7^2-2\cdot2)^2-2\cdot2^2=\\ \\ =(49-4)^2-2\cdot4=45^2-8=2025-8=2017$

Answer 2

To find the value of X1^4 + X2^4, we can use the following steps:

First, we can use the sum and product of roots formula to find the values of X1 and X2. According to this formula, if the roots of the quadratic equation ax^2 + bx + c = 0 are X1 and X2, then:

X1 + X2 = -b/a X1*X2 = c/a

In this case, the given equation is:

X1 + X2 = 7 X1*X2 = 2

Using these equations, we can solve for X1 and X2:

X1 = (7 + sqrt(7^2 - 42))/2 = (7 + sqrt(41))/2 X2 = (7 - sqrt(7^2 - 42))/2 = (7 - sqrt(41))/2

Next, we can use the identity (X^2 + Y^2)^2 - 2(X^4 + Y^4) = (X-Y)^4 to find the value of X1^4 + X2^4. Using this identity, we get:

(X1^2 + X2^2)^2 - 2(X1^4 + X2^4) = (X1 - X2)^4

We can find the values of X1^2 and X2^2 using the equations:

X1^2 + X2^2 = (X1 + X2)^2 - 2X1X2 = 7^2 - 22 = 45

Substituting the values of X1 and X2, we get:

(X1^2 + X2^2)^2 - 2(X1^4 + X2^4) = (X1 - X2)^4 = (sqrt(41))^4 = 1681

Simplifying the left-hand side, we get:

(X1^2 + X2^2)^2 - 2(X1^4 + X2^4) = 45^2 - 2(X1^4 + X2^4) = 1681

Solving for X1^4 + X2^4, we get:

X1^4 + X2^4 = (45^2 - 1681)/(-2) = 83

Therefore, the value of X1^4 + X2^4 is 83.