Solving the Equation x2 - 7xy 12y2 0 for x/y

In this article, we will solve the equation x2 - 7xy 12y2 0 for the ratio x/y. We will explore several methods to find the solution and discuss the algebraic manipulations involved. This process is crucial for understanding how to solve quadratic equations and manipulate algebraic expressions.

We begin by dividing each term by xy

Start with the equation:

x2 - 7xy 12y2 0

Divide every term by xy to simplify:

x2/xy - 7xy/xy 12y2/xy 0

which simplifies to:

x/y - 7 12/y * x/y 0

Let x/y a

Substitute x/y a into the equation:

a - 7 12/a 0

Multiply through by a to clear the fraction:

a2 - 7a 12 0

Factorize the quadratic equation:

(a - 4) * (a - 3) 0

The solutions are:

a 4, a 3

The value of x/y is 4 or 3

Hence, x/y 4 or x/y 3.

A second method: Completing the square

Alternatively, we can solve the equation using the method of completing the square:

x2 - 7xy 12y2 0

Let's define z x/y. Substitute:

z2 - 7z 12 0

Complete the square:

(z - 7/2)2 - 49/4 12 0

Clean up the constant term:

(z - 7/2)2 - 1/4 0

Solve for z:

(z - 7/2) - 1/2 0

This gives:

z 4, z 3

Hence, x/y 4 or x/y 3.

A third method: Substitution and factoring

Another approach is to rewrite the equation in terms of x/y and then solve it:

x2 - 7xy 12y2 0

Let x/y p. Then y x/p, and the equation becomes:

x2 - 7x * (x/p) 12(x/p)2 0

Multiply through by p2 to clear the fractions:

x2 * p2 - 7x2 * p/1 12x2 * p-2 0

This simplifies to a quadratic in p:

p2 - 7p 12 0

Factorize the quadratic equation:

(p - 4) * (p - 3) 0

The solutions are:

p 4, p 3

Hence, x/y 4 or x/y 3.

Conclusion: The value of x/y in the equation x2 - 7xy 12y2 0 can be determined through various methods such as dividing by xy, completing the square, or substitution and factoring. The solutions are x/y 4 or x/y 3.