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Chapter 9: Problem 25

Write the expression in factored form. $$ (x-4)^{2}-4 $$

Short Answer

Expert verified
Question: Determine the factored form of the expression \((x-4)^2 - 4\). Answer: \((x-2)(x-6)\)

Step by step solution

01

Identify \(a\) and \(b\).

In our expression, \((x-4)^2\) will be \(a^2\) and \(4\) will be \(b^2\). Therefore, we have \(a = (x-4)\) and \(b = 2\).
02

Apply the difference of squares formula

Using the values of \(a\) and \(b\) that we found, we apply the formula: \(a^2 - b^2 = (a+b)(a-b)\). In our case, \((x-4)^2 - 4 = ((x-4) + 2)((x-4) - 2)\).
03

Simplify the factored expression

Now we simply need to simplify each part of the factored expression. We have: \(((x-4) + 2)((x-4) - 2) = (x - 2)(x - 6)\). Thus the factored form of the given expression is: \((x-2)(x-6)\).

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Key Concepts

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

Difference of Squares
The difference of squares is a fundamental concept in algebra that simplifies expressions where two squares are subtracted. In our given exercise, we had \((x-4)^2 - 4\), which can be categorized as a difference of squares because it fits the form \(a^2 - b^2\). This technique relies on recognizing the expression as the difference between two perfect squares.Understanding the Formula
  • The general formula for the difference of squares is: \(a^2 - b^2 = (a+b)(a-b)\).
  • It breaks down a subtraction between squares into a product of two binomials.
  • This is very useful for simplifying complex algebraic expressions.
To apply this to our exercise, we identify \(a\) as \(x-4\) and \(b\) as \2\. By substituting these into our formula, we can efficiently factor the expression into its simpler components.Recognizing patterns like these can save time and simplify computation in algebra, making it easier to solve equations.
Factored Form
In algebra, the factored form of an expression represents it as a product of its factors. It is often more concise and easier to understand. For example, \(x^2 - 6x + 8\) can be factored into \ (x-2)(x-4) \, which means finding the numbers that multiply together to form the given expression.Why Factoring is Useful
  • Factoring can make it easier to solve linear and quadratic equations.
  • It helps in analyzing the properties of equations, like intercepts and roots.
  • It simplifies complex expressions and can aid in number theory problems.
Factoring can also confirm equality by expanding the factors and checking if they equal the original expression. It is crucial in different stages of algebra, ensuring a deep understanding of expressions and assisting with simplification in higher math.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the basis of algebra and are used to express associative, commutative, and distributive properties.Components of Algebraic Expressions
  • Variables: Symbols used to represent unknown values. In our case, \(x\) is the variable.
  • Constants: Known numbers like \(4\), which remain unchanged.
  • Operations: Include addition, subtraction, multiplication, and division.
  • Terms: Components of an expression, like \((x-4)^2\) and \(4\) in our exercise.
Understanding these components is key to manipulating and simplifying algebraic expressions. The goal is often to factor, expand, or rewrite expressions to solve equations or make calculations more manageable. Algebraic expressions are foundational in understanding more complex mathematical concepts and are applied in various fields such as engineering, economics, and science.

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Most popular questions from this chapter

We know that if \(A \cdot B=0,\) then either \(A=0\) or \(B=0 .\) If \(A \cdot B=6,\) does that imply that either \(A=6\) or \(B=6\) ? Explain your answer.

Solve by (a) Completing the square (b) Using the quadratic formula $$ x^{2}-22 x+10=0 $$

If a diver jumps off a diving board that is \(6 \mathrm{ft}\) above the water at a velocity of \(20 \mathrm{ft} / \mathrm{sec},\) his height, \(s,\) in feet, above the water can be modeled by \(s(t)=-16 t^{2}+\) \(20 t+6,\) where \(t \geq 0\) is in seconds. (a) How long is the diver in the air before he hits the water? (b) What is the maximum height achieved and when does it occur?

The stopping distance, in feet, of a car traveling at \(v\) miles per hour is given by \(^{5}\) $$ d=2.2 v+\frac{v^{2}}{20} $$ (a) What is the stopping distance of a car going 30 mph? 60 mph? 90 mph? (b) If the stopping distance of a car is 500 feet, use a graph to determine how fast it was going when it braked, and check your answer using the quadratic formula.

Find the vertex of the parabola. $$ y=x^{2}-2 x+3 $$
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