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Magazine Chapter 0: Problem 9
\((2 x-3)^{2}\)
Short Answer
Expert verified
The simplified expression of \((2x-3)^2\) is \(4x^2 -12x + 9\).
Step by step solution
01
Write down the given expression
The given expression is \((2x-3)^2\).
02
Expand the expression using distributive property (FOIL method)
To expand \((2x-3)^2\), we have to multiply the binomial (2x-3) by itself. Using the FOIL method, we need to multiply the First terms, the Outer terms, the Inner terms, and the Last terms.
\((2x-3)(2x-3)\)
03
Multiply the First terms
First, we will multiply the First terms, which are "2x" from both binomials.
\(2x \cdot 2x = 4x^2\)
04
Multiply the Outer terms
Next, we will multiply the Outer terms, which are "2x" from the first binomial and "-3" from the second binomial.
\(2x \cdot -3 = -6x\)
05
Multiply the Inner terms
Now, we will multiply the Inner terms, which are "-3" from the first binomial and "2x" from the second binomial.
\(-3 \cdot 2x = -6x\)
06
Multiply the Last terms
Lastly, we will multiply the Last terms, which are "-3" from both binomials.
\(-3 \cdot -3 = 9\)
07
Combine all terms
Now that we have all the terms, we will combine them to find the expanded form of the given expression.
\(4x^2 -6x -6x + 9\)
08
Simplify by combining like terms
We can simplify the expanded expression by combining like terms.
\(4x^2 -12x + 9\)
The simplified expression is \(4x^2 -12x + 9\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Theorem
The Binomial Theorem helps us expand expressions that are raised to a power, like \((a+b)^n\). This theorem enables us to simplify complex binomial expansions without repeatedly multiplying the binomial by itself.
It provides a quick way to expand expressions like \((2x-3)^2\) by using a formula that involves coefficients known as binomial coefficients.
It provides a quick way to expand expressions like \((2x-3)^2\) by using a formula that involves coefficients known as binomial coefficients.
- The pattern it follows is: \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
- Here, \(\binom{n}{k}\) are the binomial coefficients, which are found using combinations.
FOIL Method
The FOIL Method is a handy technique for multiplying two binomials.
In our exercise, the expression \((2x-3)(2x-3)\) used FOIL to expand: this involves four steps:
In our exercise, the expression \((2x-3)(2x-3)\) used FOIL to expand: this involves four steps:
- First: Multiply the first terms from each binomial: "2x" and "2x," resulting in \(4x^2\).
- Outer: Multiply the outer terms, which are "2x" and "-3," resulting in \(-6x\).
- Inner: Multiply the inner terms, "-3" and "2x," which gives another \(-6x\).
- Last: Multiply the last terms, "-3" and "-3," equaling \(+9\).
Distributive Property
The Distributive Property is a fundamental principle in algebra, used to multiply one term across a sum or difference. It states: \(a(b + c) = ab + ac\).
This property formed the backbone of our exercise by allowing the multiplication of each term inside a binomial with every term in another binomial.
This property formed the backbone of our exercise by allowing the multiplication of each term inside a binomial with every term in another binomial.
- When expanding \((2x-3)(2x-3)\), the distributive property ensures each term in the first binomial multiplies every term in the second binomial.
- This results in combining results from FOIL, such as in the steps where we calculated \(4x^2, -6x, -6x,\) and \(+9\).
