Factorise:
$2 x^2-7 x y+6 y^2$
The objective is to factorize the quadratic expression \(2x^2 - 7xy + 6y^2\). This is a homogeneous quadratic trinomial in two variables \(x\) and \(y\), which can be written in the general form \(ax^2 + bxy + cy^2\), where \(a=2\), \(b=-7\), and \(c=6\). To factorize such a trinomial, we look for two linear factors of the form \((px + qy)(rx + sy)\).
We use the 'ac-method' to split the middle term. We need to find two numbers whose product is \(a \times c = 2 \times 6 = 12\) and whose sum is \(b = -7\). The factors of 12 include pairs like (1, 12), (2, 6), (3, 4), (-1, -12), (-2, -6), and (-3, -4). Among these, the pair \(-3\) and \(-4\) sums to \(-3 + (-4) = -7\). Thus, we can split the middle term \(-7xy\) into \(-3xy - 4xy\).
Now, we rewrite the original expression by substituting the split middle term:
\[2x^2 - 3xy - 4xy + 6y^2\]
Next, we group the terms:
\[(2x^2 - 3xy) + (-4xy + 6y^2)\]
Now, we factor out the greatest common factor from each group:
Substituting these back, we have:
\[x(2x - 3y) - 2y(2x - 3y)\]
Since the binomial \((2x - 3y)\) is common to both terms, we can factor it out:
\[(x - 2y)(2x - 3y)\]
Thus, the factorization of \(2x^2 - 7xy + 6y^2\) is \((x - 2y)(2x - 3y)\).
Factoring quadratic expressions is fundamental in many practical areas. For example, if you are designing a physical layout, such as a garden or a building floor, and the area is represented by a quadratic expression like \(2x^2 - 7xy + 6y^2\), finding the linear factors \((x - 2y)\) and \((2x - 3y)\) helps you determine the dimensions (length and width) of that space. This is a common task in architecture, engineering, and resource planning, where understanding the components of a system allows you to optimize material usage and space allocation.