Question

Solve the equation 9x²y² - 12xy + 4 = 0, expressing y in terms of x.

Answer 1

Explanation:

We can solve the given equation for y in terms of x by treating it as a quadratic equation in y. To do so, we can use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

In this case, we can rearrange the equation to get:

9x^2y^2 - 12xy + 4 = 0

which can be written as:

(3xy)^2 - 2(3xy)(2) + 2^2 - 2^2 = 0

This is a quadratic equation in 3xy, which can be solved using the quadratic formula:

3xy = [2 ± sqrt(2^2 - 4(1)(-2^2))]/(2*1)

3xy = [2 ± sqrt(4 + 32)]/2

3xy = [2 ± 2sqrt(9)]/2

3xy = 1 ± 3

Therefore, we have two possible solutions:

3xy = 1 + 3 = 4 or 3xy = 1 - 3 = -2

Solving for y in terms of x, we get:

3xy = 4 => y = 4/(3x)

or

3xy = -2 => y = -2/(3x)

Therefore, the solutions to the given equation are:

y = 4/(3x) or y = -2/(3x)

Answer 2

To solve the equation 9x²y² - 12xy + 4 = 0 and express y in terms of x, we can use the quadratic formula. However, the equation has no real solutions.

To solve the equation 9x²y² - 12xy + 4 = 0 and express y in terms of x, we can use the quadratic formula. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

In this equation, the coefficients are a = 9, b = -12, and c = 4. Substituting these values into the quadratic formula and simplifying, we get:

x = (6 ± √(36 - 144)) / 18

x = (6 ± √(-108)) / 18

Since the discriminant (√(b² - 4ac)) is negative, the equation has no real solutions. Therefore, we cannot express y in terms of x in this case.