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Chapter 5: Problem 88

Perform the indicated operations. $$ (5 k-6)^{2} $$

Short Answer

Expert verified
The expanded form is \(25k^2 - 60k + 36\).

Step by step solution

01

Understand the Operation

We need to expand the expression \((5k - 6)^2\). This means multiplying \((5k - 6)\) by itself.
02

Apply the FOIL Method

Use the FOIL method which stands for First, Outer, Inner, Last. Expand \((5k - 6)(5k - 6)\) as follows:- First: Multiply the first terms: \(5k \times 5k = 25k^2\).- Outer: Multiply the outer terms: \(5k \times (-6) = -30k\).- Inner: Multiply the inner terms: \(-6 \times 5k = -30k\).- Last: Multiply the last terms: \(-6 \times (-6) = 36\).
03

Combine Like Terms

Add all the resulting terms from the FOIL method: \[25k^2 - 30k - 30k + 36\] Combine the like terms \(-30k\) and \(-30k\) to get:\[25k^2 - 60k + 36\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

FOIL Method
When expanding expressions like \((5k - 6)^2\), the FOIL method is a handy tool. FOIL is an acronym for First, Outer, Inner, Last. This process helps you expand the expression systematically. First, recognize that \((5k - 6)^2 = (5k - 6)(5k - 6)\). When using the FOIL method:
  • First: Multiply the first terms of each binomial. Here, that's \(5k \times 5k = 25k^2\).
  • Outer: Then, multiply the outer terms. This involves \(5k \times (-6) = -30k\).
  • Inner: Next, multiply the inner terms, which are \(-6 \times 5k = -30k\).
  • Last: Finally, multiply the last terms. Here, it is \(-6 \times -6 = 36\).
You have now expanded the expression into separate terms using this method. The resulting expression is \(25k^2 - 30k - 30k + 36\). The FOIL method streamlines binomial multiplication by ensuring every element has been accounted for during the process.
Combining Like Terms
After applying the FOIL method, the result will include several terms: \(25k^2 - 30k - 30k + 36\). The step to "Combine Like Terms" is all about simplifying your expression further. This means looking for terms that have the same variable part and combining coefficients. In this instance:
  • Like Terms: The terms \(-30k - 30k\) both contain the \(k\) variable. Therefore, they can be combined.
  • Combining: Add the coefficients of these like terms: \(-30k - 30k = -60k\).
  • Result: After combining, the expression simplifies to \(25k^2 - 60k + 36\).
The goal of combining like terms is simplification, and it helps to reduce complexity. Whenever you see the same variables with their respective exponents, consider them like terms that might be able to be reduced further.
Quadratic Expressions
Quadratic expressions are polynomial expressions where the highest degree term is a square, denoted as \(ax^2\). In our case, the expanded and simplified expression from \((5k - 6)^2\) is \(25k^2 - 60k + 36\), which is a classic quadratic expression.
  • Structure: A standard quadratic expression has the form \(ax^2 + bx + c\).
  • Components:
    • \(a = 25\), which is the coefficient of the quadratic term.
    • \(b = -60\), which is the coefficient of the linear term.
    • \(c = 36\), which is the constant term.
  • Purpose: Quadratic expressions are critical in functions and equations that involve parabolas. They appear frequently in problems involving motion and optimization.
Understanding the composition and significance of quadratics can aid in grasping further algebraic concepts. The variables and their coefficients define the shape and properties of the parabola in a quadratic graph.

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Most popular questions from this chapter

The polynomial function \(f(x)=0.001 x^{3}-0.12 x^{2}+3.6 x+10\) models the path of a portion of the track of a roller coaster. Use the function equation to find the height of the track for \(x=0,20,40,\) and 60 (PICUTURE NOT COPY)

Graph the line with the given characteristics. Passing through \((-2,-1) ;\) slope \(-\frac{2}{3}\)

Factor. Assume all variables represent natural numbers. $$ a^{3 b}-c^{3 b} $$

Factor the polynomial in part a. Then use your answer to part a to find the remaining factorizations. (No new work is necessary!). a. \(3 a^{2}+12 a+12\) b. \(3 a^{2}+12 a b+12 b^{2}\)

Find the domain of the function defined by: $$ f(x)=\frac{3}{x^{2}-x-6} $$
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