Verified Answer :

  • Calculate the product \(A \times C = 4 \times (-18) = -72\).
  • Find two numbers that multiply to \(-72\) and add to \(-21\), which are \(-24\) and \(3\).
  • Rewrite the middle term as \(-24xy + 3xy\) and group the terms: \((4x^2 - 24xy) + (3xy - 18y^2)\).
  • Extract the common binomial \((x - 6y)\) to get the final factored form \(\boxed{(4x + 3y)(x - 6y)}\).

Explanation

  1. Analysis of the Expression

We are analyzing the quadratic expression \(4x^2 - 21xy - 18y^2\). This is a homogeneous quadratic expression of the form \(Ax^2 + Bxy + Cy^2\), where \(A = 4\), \(B = -21\), and \(C = -18\). Our goal is to factor this expression into the product of two linear binomials. To do this, we use the AC method (also known as the grouping method), which involves finding two numbers whose product is \(A \times C\) and whose sum is \(B\).

  1. Identifying the Factors for the Middle Term

First, calculate the product \(A \times C\):
\[A \times C = 4 \times (-18) = -72\]
Next, find two numbers \(p\) and \(q\) such that \(p \times q = -72\) and \(p + q = -21\).
Testing factors of \(-72\):

  • \(-1 \times 72 = -72\), sum is \(71\)
  • \(-2 \times 36 = -72\), sum is \(34\)
  • \(-3 \times 24 = -72\), sum is \(21\) (not \(-21\))
  • \(-24 \times 3 = -72\), sum is \(-21\)

We found the correct pair: \(-24\) and \(3\).

  1. Factoring by Grouping

Now, rewrite the middle term \(-21xy\) using the two numbers we identified:
\[4x^2 - 24xy + 3xy - 18y^2\]
Next, group the terms:
\[(4x^2 - 24xy) + (3xy - 18y^2)\]
Factor out the greatest common factor (GCF) from each group:

  • From the first group \((4x^2 - 24xy)\), the GCF is \(4x\): \(4x(x - 6y)\)
  • From the second group \((3xy - 18y^2)\), the GCF is \(3y\): \(3y(x - 6y)\)

Now the expression is:
\[4x(x - 6y) + 3y(x - 6y)\]
Since \((x - 6y)\) is common to both terms, we can factor it out:
\[(4x + 3y)(x - 6y)\]
Thus, the factored form is \((4x + 3y)(x - 6y)\).

Examples

Factoring quadratics is a foundational skill in algebra that allows us to simplify complex expressions and solve equations efficiently. For instance, if you are designing a rectangular garden with an area given by an expression like \(4x^2 - 21xy - 18y^2\), factoring it into \((4x + 3y)(x - 6y)\) reveals the dimensions (length and width) of the garden in terms of the variables \(x\) and \(y\). This helps in planning irrigation, fencing, and landscaping based on those dimensions.

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