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Magazine Chapter 4: Problem 52
Factor. \(36 x^{2}-60 x+25\)
Short Answer
Expert verified
The expression factors to \((6x - 5)^2\).
Step by step solution
01
Identify the quadratic form
The given expression is a quadratic expression in the form \(ax^2 + bx + c\) with \(a = 36\), \(b = -60\), and \(c = 25\).
02
Check for perfect square trinomial
A perfect square trinomial can be expressed as \((mx + n)^2\). Let's verify if this expression is a perfect square by using the formula \((m^2x^2 + 2mnp + n^2)\).
03
Compare coefficients
Using the perfect square trinomial formula, compare coefficients:- \(m^2 = 36\) implies \(m = 6\) or \(m = -6\),- \(n^2 = 25\) implies \(n = 5\) or \(n = -5\).- Check the middle term: \(2mn = -60\).
04
Verify coefficient product
Let's test \(m = 6\) and \(n = -5\): - \(2 \cdot 6 \cdot (-5) = -60\) which matches the middle term.Thus, \(m = 6\) and \(n = -5\) satisfy all conditions.
05
Write the factorized form
Since we verified the perfect square trinomial, the factorized form of \(36x^2 - 60x + 25\) is \((6x - 5)^2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Perfect Square Trinomial
When working with quadratics, a **perfect square trinomial** is a special kind of expression.
It takes the form \((mx + n)^2 = m^2x^2 + 2mnx + n^2\).
This is particularly useful because it enables us to rewrite the trinomial as a squared binomial, simplifying the expression significantly. In our specific exercise, the expression is \(36x^2 - 60x + 25\).
Recognizing such patterns is important in simplifying polynomial expressions quickly.Identifying a perfect square trinomial involves:
It takes the form \((mx + n)^2 = m^2x^2 + 2mnx + n^2\).
This is particularly useful because it enables us to rewrite the trinomial as a squared binomial, simplifying the expression significantly. In our specific exercise, the expression is \(36x^2 - 60x + 25\).
Recognizing such patterns is important in simplifying polynomial expressions quickly.Identifying a perfect square trinomial involves:
- Checking that each term corresponds to the formula \(m^2x^2 + 2mnx + n^2\).
- Ensuring the first and last terms are perfect squares, i.e., the square roots are whole numbers or rational numbers.
- Verifying that the middle term fits \(2mnx\).
Quadratic Expression
A **quadratic expression** is a polynomial where the highest power of the variable is two.
The standard form of a quadratic expression is \(ax^2 + bx + c\).
In our exercise, the quadratic expression is \(36x^2 - 60x + 25\), with \(a = 36\), \(b = -60\), and \(c = 25\).Quadratic expressions often appear in various mathematical contexts like physics, engineering, and economics.
In solving quadratic expressions, especially when factoring, the key steps involve:
The standard form of a quadratic expression is \(ax^2 + bx + c\).
In our exercise, the quadratic expression is \(36x^2 - 60x + 25\), with \(a = 36\), \(b = -60\), and \(c = 25\).Quadratic expressions often appear in various mathematical contexts like physics, engineering, and economics.
In solving quadratic expressions, especially when factoring, the key steps involve:
- Recognizing the structure \(ax^2 + bx + c\).
- Identifying if it forms any recognizable patterns, such as a perfect square trinomial.
- Using factorization techniques, if applicable, to simplify or solve the equation.
Coefficient Comparison
In mathematics, **coefficient comparison** is a technique used to match the components of an equation to identify unknown values.
When factoring quadratics, this method is crucial, especially when determining if an expression is a perfect square trinomial.In our example, \(36x^2 - 60x + 25\), we compared coefficients as follows:
When factoring quadratics, this method is crucial, especially when determining if an expression is a perfect square trinomial.In our example, \(36x^2 - 60x + 25\), we compared coefficients as follows:
- Determine \(m\) from \(m^2 = 36\).
We learn that \(m = 6\) or \(m = -6\). - Find \(n\) where \(n^2 = 25\).
This gives us \(n = 5\) or \(n = -5\). - Verify the middle term through \(2mn = -60\).
This confirms the pairings.
Factorization
**Factorization** is the process of breaking down an expression into a product of simpler expressions, or factors.
It is a fundamental skill in algebra, aiding in solving polynomial equations efficiently.The factorization of a quadratic expression like \(36x^2 - 60x + 25\) involves recognizing structural patterns.
For perfect square trinomials, this means confirming the settings for \((mx + n)^2\), thus making the trinomial factored format clear.The steps generally include:
It is a fundamental skill in algebra, aiding in solving polynomial equations efficiently.The factorization of a quadratic expression like \(36x^2 - 60x + 25\) involves recognizing structural patterns.
For perfect square trinomials, this means confirming the settings for \((mx + n)^2\), thus making the trinomial factored format clear.The steps generally include:
- Identifying recognizable patterns, such as perfect square trinomials or simple quadratic factors.
- Applying methods like \(\text{(GCF, Difference of Squares, or Completing the Square)}\) as appropriate.
- Checking the result by expanding or verifying with initial conditions.
