Question

(3x+1)^(3)=27x^(2)(x+1)

Answer 1

$(3x+1)^3=27x^2(x+1)\\27x^3+27x^2+9x+1=27x^3+27x^2\\9x+1=0\\9x=-1\\x=-\frac{1}{9} =-0,1$

Answer 2

Let's solve the equation step-by-step:

Step 1: Expand the left side of the equation using the binomial theorem. The binomial theorem states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where the summation is taken from k = 0 to n, and (n choose k) = n! / (k! * (n-k)!).

In this case, we have (3x + 1)^3 = Σ (3 choose k) * (3x)^(3-k) * (1^k) from k = 0 to 3.

(3x + 1)^3 = (3 choose 0) * (3x)^(3-0) * (1^0) + (3 choose 1) * (3x)^(3-1) * (1^1) + (3 choose 2) * (3x)^(3-2) * (1^2) + (3 choose 3) * (3x)^(3-3) * (1^3)

(3x + 1)^3 = (1) * (3x)^3 * (1) + (3) * (3x)^2 * (1) + (3) * (3x) * (1^2) + (1) * (1^3) (3x + 1)^3 = 27x^3 + 27x^2 + 9x + 1

Step 2: Simplify the right side of the equation.

27x^2 * (x + 1) = 27x^3 + 27x^2

Step 3: Equate the left and right sides of the equation.

Now we have the equation: 27x^3 + 27x^2 + 9x + 1 = 27x^3 + 27x^2

Step 4: Solve for x.

To solve for x, we need to isolate x on one side of the equation. Let's move all the terms involving x to the left side:

27x^3 + 27x^2 + 9x + 1 - (27x^3 + 27x^2) = 0

Simplify the equation:

9x + 1 = 0

Subtract 1 from both sides:

9x = -1

Finally, divide both sides by 9 to get the value of x:

x = -1/9

So, the solution to the equation (3x + 1)^3 = 27x^2 * (x + 1) is x = -1/9.