Find the Perfect Square Trinomial (2x-3)(3x-3)
This problem asks you to perform a multiplication of two binomials and then identify and present the resulting expression as a perfect square trinomial. A perfect square trinomial is an algebraic expression that is the square of a binomial - in other words, it takes the form (ax ± b)². The task involves applying foil (First, Outer, Inner, Last) method or distributive property to expand and simplify the product of the given binomials (2x-3)(3x-3) and then rewrite it in the form of a squared binomial if it does indeed represent a perfect square.
$\left(\right. 2 x - 3 \left.\right) \left(\right. 3 x - 3 \left.\right)$
Answer
Solution:
Step 1: Expand the Expression
Use the FOIL (First, Outer, Inner, Last) method to expand $(2x-3)(3x-3)$.
Step 1.1: Distribute the First Terms
Multiply the first terms of each binomial: $2x(3x-3)$ and then distribute $-3(3x-3)$.
Step 1.2: Continue Distribution
Continue distributing: $2x(3x) + 2x(-3) - 3(3x-3)$.
Step 1.3: Complete Distribution
Finish distributing: $2x(3x) + 2x(-3) - 3(3x) - 3(-3)$.
Step 2: Combine Like Terms
Simplify the expression by combining like terms.
Step 2.1: Simplify Each Term
Simplify each distributed term individually.
Step 2.1.1: Apply Commutative Property
Use the commutative property to rearrange the multiplication: $2 \cdot 3x^2 + 2x(-3) - 3(3x) - 3(-3)$.
Step 2.1.2: Multiply Exponents
Multiply terms with exponents by adding their powers.
Step 2.1.2.1: Rearrange Exponents
Rearrange the terms with exponents: $2 \cdot 3(x \cdot x) + 2x(-3) - 3(3x) - 3(-3)$.
Step 2.1.2.2: Simplify Exponents
Simplify the exponents: $2 \cdot 3x^2 + 2x(-3) - 3(3x) - 3(-3)$.
Step 2.1.3: Multiply Constants
Multiply the constants: $6x^2 + 2x(-3) - 3(3x) - 3(-3)$.
Step 2.1.4: Multiply by Negative Constants
Multiply by the negative constants: $6x^2 - 6x - 3(3x) - 3(-3)$.
Step 2.1.5: Multiply Remaining Terms
Multiply the remaining terms: $6x^2 - 6x - 9x - 3(-3)$.
Step 2.1.6: Multiply Negative by Negative
Multiply the negative terms: $6x^2 - 6x - 9x + 9$.
Step 2.2: Combine Like Terms
Combine the like terms to simplify further: $6x^2 - 15x + 9$.
Knowledge Notes:
To solve this problem, we used the following knowledge points:
FOIL Method: This is a technique used to expand the product of two binomials. FOIL stands for First, Outer, Inner, Last, referring to the terms in each binomial that are multiplied together.
Distributive Property: This property states that $a(b + c) = ab + ac$. It allows us to multiply a single term by each term inside a parenthesis.
Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not change the product, i.e., $ab = ba$.
Combining Like Terms: Terms that have the same variable raised to the same power can be combined by adding or subtracting their coefficients.
Multiplying Exponents: When multiplying terms with the same base, you add the exponents, as in $x^a \cdot x^b = x^{a+b}$.
Multiplying Negative Numbers: When multiplying two negative numbers, the result is positive, as in $(-a)(-b) = ab$.
By applying these principles, we expanded and simplified the expression $(2x-3)(3x-3)$ to find the perfect square trinomial $6x^2 - 15x + 9$.
