Question

Решите пожалуйста 1.Решите уравнения методом замены1.1 3(x^2+4/x^2)+2(x+2/x)=211.2 2(x^2+9/x^2)-3(x-3/x)=142. Для каждого значения параметра а решите уравнение2.1 ((x^2-(2+3a)x+6a))/(x^2-9)2.2 ((x^2+(1+2a)x+2a))/(x^2-4)

Answer 1

Ответ:

Объяснение:

1)  3(x^2+4/x^2)+2(x+2/x)=21    2)

x^3+4x^2-21x=0

x(x^2+4x-21)=0

x(x+7)(x-3)=0

x1=0

x2=-7

x3=3

Answer 2

1.1 To solve the equation using the method of substitution, we let u = x + 1/x. Then, u^2 = (x + 1/x)^2 = x^2 + 2 + 1/x^2, and u^2 - 2 = x^2 + 1/x^2.

Now, the equation becomes:

3(u^2 - 2) + 2u = 211.

Simplify the equation:

3u^2 + 2u - 217 = 0.

Now, we have a quadratic equation in terms of u. We can solve it using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:

u = (-b ± √(b^2 - 4ac)) / 2a,

where a = 3, b = 2, and c = -217.

u = (-2 ± √(2^2 - 4 * 3 * (-217))) / 2 * 3 u = (-2 ± √(4 + 2604)) / 6 u = (-2 ± √2608) / 6 u = (-2 ± 51.07) / 6.

Now we have two possible values for u:

u₁ = (51.07 - 2) / 6 ≈ 8.512, u₂ = (-51.07 - 2) / 6 ≈ -9.179.

Now, we need to find the corresponding values of x:

For u₁:

x + 1/x = 8.512.

We solve this quadratic equation to find x:

x^2 + 1 = 8.512x x^2 - 8.512x + 1 = 0.

Using the quadratic formula:

x = (8.512 ± √(8.512^2 - 4 * 1 * 1)) / 2 * 1 x = (8.512 ± √(72.310144 - 4)) / 2 x = (8.512 ± √68.310144) / 2 x ≈ (8.512 ± 8.273) / 2.

So the two solutions for x are:

x₁ ≈ (8.512 + 8.273) / 2 ≈ 8.393, x₂ ≈ (8.512 - 8.273) / 2 ≈ 0.120.

For u₂:

x + 1/x = -9.179.

We solve this quadratic equation to find x:

x^2 + 1 = -9.179x x^2 + 9.179x + 1 = 0.

Using the quadratic formula:

x = (-9.179 ± √(9.179^2 - 4 * 1 * 1)) / 2 * 1 x = (-9.179 ± √84.210241 - 4) / 2 x = (-9.179 ± √80.210241) / 2 x ≈ (-9.179 ± 8.955) / 2.

So the two solutions for x are:

x₃ ≈ (-9.179 + 8.955) / 2 ≈ -0.112, x₄ ≈ (-9.179 - 8.955) / 2 ≈ -8.567.

Therefore, the solutions to the original equation are:

x₁ ≈ 8.393, x₂ ≈ 0.120, x₃ ≈ -0.112, x₄ ≈ -8.567.

1.2 To solve the equation using the method of substitution, let u = x + 1/x. Then, u^2 = (x + 1/x)^2 = x^2 + 2 + 1/x^2, and u^2 - 2 = x^2 + 1/x^2.

Now, the equation becomes:

2(u^2 - 2) - 3u = 14.

Simplify the equation:

2u^2 - 4 - 3u = 14.

Move all terms to one side:

2u^2 - 3u - 18 = 0.

Now, we have a quadratic equation in terms of u. We can solve it using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:

u = (-b ± √(b^2 - 4ac)) / 2a,

where a = 2, b = -3, and c = -18.

u = (3 ± √((-3)^2 - 4 * 2 * (-18))) / 2 * 2 u = (3 ± √(9 + 144)) / 4 u = (3 ± √153) / 4.

Now we have two possible values for u:

u₁ = (3 + √153) / 4 ≈ 3.932, u₂ = (3 - √153) / 4 ≈ -2.432.

Now, we need to find the corresponding values of x:

For u₁:

x + 1/x = 3.932.

We solve this quadratic equation to find x:

x^2 + 1 = 3.932x x^2 - 3.932x + 1 = 0.

Using the quadratic formula:

x = (3.932 ± √(3.932^2 - 4 * 1 * 1)) / 2 * 1 x = (3.932 ± √(15.494624 - 4)) / 2 x = (3.932 ± √11.494624) / 2 x ≈ (3.932 ± 3.39) / 2.

So the two solutions for x are:

x₁ ≈ (3.932 + 3.39) / 2 ≈ 3.661, x₂ ≈ (3.932 - 3.39) / 2 ≈ 0.271.

For u₂:

x + 1/x = -2.432.

We solve this quadratic equation to find x:

x^2 + 1 = -2.432x x^2 + 2.432x + 1 = 0.

Using the quadratic formula:

x = (-2.432 ± √(2.432^2 - 4 * 1 * 1)) / 2 * 1 x = (-2.432 ± √5.917824 - 4) / 2 x = (-2.432 ± √1.917824) / 2 x ≈ (-2.432 ± 1.384) / 2.

So the two solutions for x are:

x₃ ≈ (-2.432 + 1.384) / 2 ≈ -0.524, x₄ ≈ (-2.432 - 1.384) / 2 ≈ -1.908.

Therefore, the solutions to the original equation are:

x₁ ≈ 3.661, x₂ ≈ 0.271