Problem 223
Question
Factor. \(49 x^{2}-28 x y+4 y^{2}\)
Step-by-Step Solution
Verified Answer
\((7x - 2y)^2\)
1Step 1: Identify the quadratic form
The given expression is in the form of a quadratic trinomial: \(49x^2 - 28xy + 4y^2\). This resembles the form \(a^2 - 2ab + b^2\), which is a perfect square trinomial.
2Step 2: Express the terms as squares
Express each term as a square: \(49x^2 = (7x)^2\) and \(4y^2 = (2y)^2\).
3Step 3: Find the middle term
The middle term, \(-28xy\), should match the product \(-2ab\). Verify: \(-2 \times 7x \times 2y = -28xy\).
4Step 4: Write the trinomial as a square
Since the trinomial matches the form \(a^2 - 2ab + b^2\), we can write it as a square: \((7x - 2y)^2\).
5Step 5: Confirm the factorization
Expand \((7x - 2y)^2\) to verify it equals the original expression: \((7x - 2y)(7x - 2y) = 49x^2 - 28xy + 4y^2\). The factorization is correct.
Key Concepts
quadratic expressionsperfect square trinomialsfactoring
quadratic expressions
Quadratic expressions are polynomial expressions of degree two. They are usually in the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.
In our example, the quadratic expression is \(49x^2 - 28xy + 4y^2\).
This particular quadratic expression has two variables, \(x\) and \(y\), making it a bit more complex than those with just one variable.
The goal of working with quadratic expressions is often to simplify, factor, or solve for the variable(s).
To factor a quadratic expression, we look for ways to rewrite it as a product of simpler expressions.
First, identify if the expression is in a recognizable form like a standard trinomial or a perfect square trinomial.
In our example, the quadratic expression is \(49x^2 - 28xy + 4y^2\).
This particular quadratic expression has two variables, \(x\) and \(y\), making it a bit more complex than those with just one variable.
The goal of working with quadratic expressions is often to simplify, factor, or solve for the variable(s).
To factor a quadratic expression, we look for ways to rewrite it as a product of simpler expressions.
First, identify if the expression is in a recognizable form like a standard trinomial or a perfect square trinomial.
perfect square trinomials
A perfect square trinomial is a specific type of quadratic expression. It follows the form \(a^2 \, + \, 2ab \, + \, b^2\) or \(a^2 \, - \, 2ab \, + \, b^2\).
These can be factored into \( (a+b)^2 \) or \((a-b)^2\), respectively.
In our problem, we have \(49x^2 - 28xy + 4y^2\), which fits the form \(a^2 - 2ab + b^2\).
We recognize \(49x^2\) as \((7x)^2\) and \(4y^2\) as \((2y)^2\).
The middle term, \(-28xy\), can be verified as \(-2(7x)(2y)\).
Thus, the expression \(49x^2 - 28xy + 4y^2\) is indeed a perfect square trinomial.
It factors neatly into \((7x - 2y)^2\).
Understanding perfect square trinomials streamlines the process of factoring complex quadratic expressions.
These can be factored into \( (a+b)^2 \) or \((a-b)^2\), respectively.
In our problem, we have \(49x^2 - 28xy + 4y^2\), which fits the form \(a^2 - 2ab + b^2\).
We recognize \(49x^2\) as \((7x)^2\) and \(4y^2\) as \((2y)^2\).
The middle term, \(-28xy\), can be verified as \(-2(7x)(2y)\).
Thus, the expression \(49x^2 - 28xy + 4y^2\) is indeed a perfect square trinomial.
It factors neatly into \((7x - 2y)^2\).
Understanding perfect square trinomials streamlines the process of factoring complex quadratic expressions.
factoring
Factoring is the process of breaking down an expression into a product of simpler expressions.
This is an essential skill for solving equations and simplifying expressions.
To factor our given expression, \(49x^2 - 28xy + 4y^2\), we first identify it as a perfect square trinomial.
We express the terms as squares: \(49x^2 = (7x)^2\) and \(4y^2 = (2y)^2\).
The middle term, \(-28xy\), confirms as \(-2(7x)(2y)\).
Thus, we write the original trinomial as \((7x - 2y)^2\).
To ensure this factorization is correct, we can expand \((7x - 2y)(7x - 2y)\), which results in \(49x^2 - 28xy + 4y^2\), matching our initial expression.
Factoring simplifies solving and working with quadratic expressions by reducing them to more manageable forms.
This is an essential skill for solving equations and simplifying expressions.
To factor our given expression, \(49x^2 - 28xy + 4y^2\), we first identify it as a perfect square trinomial.
We express the terms as squares: \(49x^2 = (7x)^2\) and \(4y^2 = (2y)^2\).
The middle term, \(-28xy\), confirms as \(-2(7x)(2y)\).
Thus, we write the original trinomial as \((7x - 2y)^2\).
To ensure this factorization is correct, we can expand \((7x - 2y)(7x - 2y)\), which results in \(49x^2 - 28xy + 4y^2\), matching our initial expression.
Factoring simplifies solving and working with quadratic expressions by reducing them to more manageable forms.