1Cademy - Factoring $$4x^2 - 12xy + 9y^2

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Factoring by Grouping

Example

Factoring $4x^2 - 12xy + 9y^2 - 25$

Factor the polynomial $4x^2 - 12xy + 9y^2 - 25$ completely. First, check for a greatest common factor. There is no overall GCF. With $4$ terms, try grouping the first $3$ terms to form a perfect square trinomial: $(4x^2 - 12xy + 9y^2) - 25$ . Factor the grouped trinomial, which fits the pattern $a^2 - 2ab + b^2$ with $a = 2x$ and $b = 3y$ . Write it as a square: $(2x - 3y)^2 - 25$ . Recognize the resulting expression as a difference of squares: $(2x - 3y)^2 - 5^2$ . Factor it as a product of conjugates: $((2x - 3y) - 5)((2x - 3y) + 5)$ . Simplify to get the completely factored form: $(2x - 3y - 5)(2x - 3y + 5)$ . Verify the result by multiplying the factors.

Updated 2026-06-03

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OpenStax Intermediate Algebra

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You are optimizing a legacy software algorithm where a material's stress tolerance is modeled by the mathematical expression ${}4x^2 - 12xy + 9y^2 - 25$ . To simplify the code, you need to rewrite this into its completely factored form. Recall the specific algebraic procedure for polynomials of this structure and arrange the factoring steps in the correct chronological order.
A structural safety inspector is using a software tool to verify the load-bearing capacity of a modular floor panel. The stress distribution is represented by the polynomial ${}4x^2 - 12xy + 9y^2 - 25$ . To perform the final calculation, the inspector must identify the completely factored form of this expression. Which of the following is the correct factored form?
An industrial design software uses the polynomial expression ${}4x^2 - 12xy + 9y^2 - 25$ to calculate the stress threshold of a mechanical part. To verify the formula's logic, a technician must identify the specific algebraic patterns encountered during the factoring process. Match each mathematical component from the expression with the correct algebraic description it represents.
A manufacturing technician is simplifying a stress-tolerance formula represented by the polynomial ${}4x^2 - 12xy + 9y^2 - 25$ . True or False: The first step in factoring this expression is to group the first three terms to form the perfect square trinomial $(2x - 3y)^2$ .
Identifying Bases in Difference of Squares
An apprentice drafting technician at an architectural firm is simplifying a land zoning boundary represented by the area formula ${}4x^2 - 12xy + 9y^2 - 25$ . To find the linear boundaries, the technician first groups the first three terms to form a perfect square trinomial, yielding the intermediate form $(2x - 3y)^2 - 25$ . To proceed with the next step of factoring, the technician must recognize that this expression matches a specific algebraic pattern.

This expression is an example of a(n) __
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