Problem 79 - College Algebra

Question

Write \(4 x^{2}-6 x y+2 y^{2}-3 x+10 y-6=0\) as a quadratic equation in \(y\) and then use the quadratic formula to express \(y\) in terms of \(x .\) Graph the resulting two equations using a graphing utility in a \([-50,70,10]\) by \([-30,50,10]\) viewing rectangle. What effect does the \(x y\) -term have on the graph of the resulting hyperbola? What problems would you encounter if you attempted to write the given equation in standard form by completing the square?

Step-by-Step Solution

Answer

The given equation, once rewritten to express y in terms of x, resulted in two equations which graphed a hyperbola. The xy term in the equation causes the hyperbola to rotate around the origin. Attempting to complete the square and convert the equation to standard form is complicated because of the xy term, which prevents the complete separation of x and y.

Step 1: Rewrite as quadratic in y

To start, rewrite the equation \(4 x^{2}-6 x y+2 y^{2}-3 x+10 y-6=0\) as a quadratic equation in \(y\) by rearranging terms: \(2y^2 - (6x)y +(4x^2 - 3x-6) = 0\)

Step 2: Solve for y using the quadratic formula

Now, recall the quadratic formula: \(x = \frac{-b\pm sqrt(b^2 - 4ac)}{2a}\). Substitute the values from our quadratic equation \(2y^2 - (6x)y +(4x^2 - 3x-6) = 0\) into the quadratic formula, treating 'y' as 'x', '6x' as 'b', '2' as 'a', and '\(4x^2 -3x-6\)' as 'c'. This yields two solutions: \(y_1 = \frac{3x + sqrt((3x)^2-4*2*(4x^2-3x-6))}{4}\) and \(y_2 = \frac{3x - sqrt((3x)^2-4*2*(4x^2-3x-6))}{4}\)

Step 3: Graph the functions

Using a graphing utility, graph the two equations within the viewing rectangle [-50, 70, 10] x [-30, 50, 10]. Observe the shape and position changes of the graph.

Step 4: Observe the effect of the xy term

Analyze the effect of the xy term on the graph. In a hyperbola, the xy term rotates the hyperbola around the origin. If the xy term is positive, the rotation is counter-clockwise. If the xy term is negative, the rotation is clockwise.

Step 5: Observe the issue in completing the square

Completing the square with this equation is problematic due to the xy term. If we attempted to put the equation in standard form by completing the square, we would have to separate the x and y terms— but because of the xy term, we cannot entirely separate x and y.