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anemone
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Find all integer solutions to the equation $4x+y+4\sqrt{xy}-28\sqrt{x}-14\sqrt{y}+48=0$.
Integer solutions refer to values of x and y that are whole numbers (positive, negative, or zero) that satisfy the given equation. In this case, the equation is $4x+y+4\sqrt{xy}-28\sqrt{x}-14\sqrt{y}+48=0$.
To find integer solutions for this equation, you can use a method called "completing the square". This involves manipulating the equation to isolate the square root terms on one side and the integer terms on the other side. Then, you can square both sides of the equation to eliminate the square roots and solve for x and y.
Yes, there are some restrictions on the values of x and y for this equation to have integer solutions. In order for the square root terms to be real numbers, the values inside the square root (xy, x, and y) must be non-negative. This means that x and y must be greater than or equal to 0, and xy must be greater than or equal to 0.
Yes, there can be multiple sets of integer solutions for this equation. This is because when you square both sides of the equation to eliminate the square root terms, you may end up with a quadratic equation that has two possible solutions for each variable.
To check if your solutions are correct, you can substitute the values of x and y into the original equation and see if it satisfies the equation. If the equation holds true, then your solutions are correct. Additionally, you can also graph the equation and see if the coordinates of your solutions lie on the graph.