Xy+3=x?x^3
To solve the equation xy + 3 = x * x^3, we need to first simplify it by combining like terms.
The term x * x^3 can be simplified to x^4 because multiplying two powers with the same base involves adding their exponents.
So the equation becomes xy + 3 = x^4.
To isolate the variable y, we need to move all terms containing y to one side of the equation.
Subtracting x^4 from both sides, we get xy - x^4 + 3 = 0.
Next, we need to factor out the common variable y.
The equation can be rewritten as y(x - x^3) + 3 = 0.
To solve for y, we can set the expression inside the parentheses equal to zero.
Thus, x - x^3 = 0.
Now, we have a quadratic equation in terms of x. To solve it, we can factor it as follows:
x(x^2 - 1) = 0.
This equation is satisfied when x equals 0 or when x^2 - 1 equals 0.
Considering the first case, if x = 0, then the equation y(x - x^3) + 3 = 0 becomes 0 + 3 = 0, which is not true.
Moving on to the second case, when x^2 - 1 = 0, we have two solutions: x = 1 and x = -1.
For x = 1, we substitute it back into the expression y(x - x^3) + 3 = 0: y(1 - 1^3) + 3 = 0. Simplifying it gives 3 = 0, which is not true.
For x = -1, we substitute it back into the expression y(x - x^3) + 3 = 0: y(-1 - (-1)^3) + 3 = 0. Simplifying it gives 0 = 0, which is true.
Therefore, the solution to the equation xy + 3 = x * x^3 is y = 0 when x = -1.