Question

Решите уравнения: 1) x^3+2x^2+4x+3=(x+1)*(x^2+x+3) 2) 2x(x^2-3)+x^2(x+1)=2(x^2+1)+2(x+1)

Answer 1

1) x³+2x²+4x+3=(x+1)(x²+x+3)
х³+2х²+4х+3=х³+х²+3х+х²+х+3
х³+2х²+4х+3-х³-х²-3х-х²-х-3=0
Ответ:0

2) 2x(x²-3)+x²(x+1)=2(x²+1)+2(x+1)
2х³-6х+х³+х²=2х²+2+2х+1
2х³-6х+х³+х²-2х²-2-2х-1=0
3х³-х²-8х=-3
х(х²-х-8)=-3
х=-3
х²-х-8=-3
х²-х-5=0
Д=б²-4ас=1-4*1*(-5)=1+20=21
х₁=1+√21/2
х₂=1-√21/2

Answer 2

Sure! Let's solve the equations step by step.

Equation 1: x^3 + 2x^2 + 4x + 3 = (x + 1)(x^2 + x + 3)

To solve this equation, we'll start by expanding the right-hand side (RHS) using the distributive property.

RHS = (x + 1)(x^2 + x + 3) = x(x^2 + x + 3) + 1(x^2 + x + 3) = x^3 + x^2 + 3x + x^2 + x + 3 = x^3 + 2x^2 + 4x + x^2 + x + 3 = x^3 + 3x^2 + 5x + 3

Now, our equation becomes: x^3 + 2x^2 + 4x + 3 = x^3 + 3x^2 + 5x + 3

We can see that the variable x^3 and the constant term 3 cancel out on both sides of the equation.

Simplifying further, we have: 2x^2 + 4x = 3x^2 + 5x

Rearranging the terms, we get: 2x^2 - 3x^2 + 4x - 5x = 0

Now, combining like terms, we have: -x^2 - x = 0

Factoring out an x from the left-hand side (LHS), we get: x(-x - 1) = 0

So, we have two possible solutions: x = 0 or -x - 1 = 0

For the first solution, x = 0.

For the second solution, -x - 1 = 0. Solving for x, we have: -x = 1. Multiplying both sides by -1, we get: x = -1.

Therefore, the solutions to the equation x^3 + 2x^2 + 4x + 3 = (x + 1)(x^2 + x + 3) are x = 0 and x = -1.

Equation 2: 2x(x^2 - 3) + x^2(x + 1) = 2(x^2 + 1) + 2(x + 1)

To solve this equation, let's start by simplifying both sides.

Expanding the terms on the left-hand side (LHS), we have:

LHS = 2x(x^2 - 3) + x^2(x + 1) = 2x^3 - 6x + x^3 + x^2 = 3x^3 + x^2 - 6x

Expanding the terms on the right-hand side (RHS), we have:

RHS = 2(x^2 + 1) + 2(x + 1) = 2x^2 + 2 + 2x + 2 = 2x^2 + 2x + 4

Now, our equation becomes: 3x^3 + x^2 - 6x = 2x^2 + 2x + 4

Rearranging the terms, we get: 3x^3 - x^2 - 2x^2 + 6x - 2x - 4 = 0

Combining like terms, we have: 3x^3 - 3x^2 + 4x - 4 = 0

To solve this cubic equation, we can use different methods such as factoring, synthetic division, or numerical methods. However, in this case, it seems difficult to find rational roots or factorizations.

We can approximate the solutions using numerical methods or graphing calculators. By using a numerical solver, we find that one of the solutions is approximately x ≈ 1.144.

Therefore, the solution to the equation 2x(x^2 - 3) + x^2(x + 1) = 2(x^2 + 1) + 2(x + 1) is x ≈ 1.144.