Use the quadratic formula to solve the equation for x in terms of y and y in terms of x. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)
2x^2 + 7xy + 1 − y^2 = 0
y in terms of x
y =
To solve the equation 2x^2 + 7xy + 1 - y^2 = 0 for y in terms of x using the quadratic formula, we can consider the equation as a quadratic equation in y. Rearrange the equation to have all the terms on one side:
y^2 + 7xy - 2x^2 - 1 = 0
Now, we can apply the quadratic formula, which states that for a quadratic equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 7x, and c = -2x^2 - 1. Substituting these values into the quadratic formula, we get:
y = (-7x ± √((7x)^2 - 4(1)(-2x^2 - 1))) / (2)
Simplifying further:
y = (-7x ± √(49x^2 + 8x^2 + 4)) / 2
y = (-7x ± √(57x^2 + 4)) / 2
Therefore, when solving the equation 2x^2 + 7xy + 1 - y^2 = 0 for y in terms of x using the quadratic formula, the solutions are:
y = (-7x + √(57x^2 + 4)) / 2, (-7x - √(57x^2 + 4)) / 2