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Chapter 8: Problem 126

APPLICATIONS The area of the checkerboard is represented by the expression \(25 x^{2}-40 x+16\) Use factoring to find the length of each side. (CHECKERBOARD NOT COPY)

Short Answer

Expert verified
The side length is \(5x - 4\).

Step by step solution

01

Understanding the Problem

The area of a square is given by the expression \(25x^2 - 40x + 16\). Our task is to factor this expression to find the length of one side since the area of a square is \( s^2 \) where \( s \) is the side length.
02

Express as a Perfect Square

Recognize that the expression \(25x^2 - 40x + 16\) is in a form that might represent a perfect square. A perfect square trinomial has the form \((ax - b)^2 = a^2x^2 - 2abx + b^2\).
03

Identify Coefficients

Compare the given expression \(25x^2 - 40x + 16\) to the perfect square form. Here: - \(a^2 = 25\), so \(a = 5\)- \(b^2 = 16\), so \(b = 4\)- Check if \(2ab = 40\). Calculation: \(2 \times 5 \times 4 = 40\).All conditions are satisfied for the perfect square.
04

Factor the Expression

Since the given expression is a perfect square as \( (5x - 4)^2 \), factor it as such.
05

Solve for the Side Length

Since the expression is factored as \((5x - 4)^2\), the side of the square is \(5x - 4\).

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Key Concepts

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

Quadratic Expressions
Quadratic expressions are a fundamental concept in algebra, often taking the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. These expressions are called 'quadratic' because the term with the highest power is squared. Quadratic expressions appear in various real-world scenarios, making them essential for students to understand.
  • The term \(ax^2\) is the quadratic term, which indicates the presence of squaring (second power).
  • The coefficient \(b\) is linked with the linear term \(x\), which affects the expression's slope.
  • The constant term \(c\) shifts the graph of the parabola up or down.
When it comes to solving these expressions, factoring is a popular method, allowing us to find solutions or simplify expressions easily. In the context of the checkerboard problem, the expression \(25x^2 - 40x + 16\) is a quadratic, indicating the need to understand such forms to interpret real-world problems correctly.
Perfect Square Trinomials
A perfect square trinomial is a special form of quadratic expression. It is the result of squaring a binomial, and it follows the structure \((ax + b)^2\). Recognizing perfect square trinomials is incredibly useful when factoring polynomials.
  • The general form of a perfect square trinomial is \(a^2x^2 + 2abx + b^2\).
  • It simplifies to \((ax + b)^2\) or \((ax - b)^2\) based on the sign of \(b\).
  • To identify a perfect square trinomial, check if both coefficient relations \(a^2\) and \(b^2\) fit, also confirm that \(2ab\) equals the middle term.
For the checkerboard problem, the quadratic \(25x^2 - 40x + 16\) fits this structure as it can be factored into \((5x - 4)^2\). Recognizing and factoring it as a perfect square allows us to find the sides of the square shape it represents, which is critical in solving geometric problems.
Algebraic Applications
Algebraic applications extend the understanding of concepts like quadratic expressions and perfect square trinomials to solve real-world problems. Recognizing these forms is not just about rewriting expressions; it helps in interpreting complex scenarios easily.
  • Algebra is a tool for converting real-world situations into mathematical models.
  • In geometric problems, expressions representing areas or volumes often lead to quadratic forms.
  • Factoring these quadratics into products of binomials gives concrete solutions to lengths and dimensions.
For example, in this checkerboard problem, the factored expression \((5x - 4)^2\) is used to determine the side length of a square. This practical connection between algebra and geometry demonstrates how important it is for students to master these skills, enabling them to solve problems across various topics.

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Most popular questions from this chapter

Perform the operations and simplify. $$\frac{2 x^{2}-2 x-4}{x^{2}+2 x-8} \cdot \frac{3 x^{2}+15 x}{x+1} \div \frac{100-4 x^{2}}{x^{2}-x-20}$$

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Passes through ( \(54,0\) ), perpendicular to the graph of \(g(x)=9 x+5\)

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Rain Gutters. A rectangular sheet of metal will be used to make a rain gutter by bending up its sides, as shown. If the ends are covered, the capacity \(f(x)\) of the gutter is a polynomial function of \(x: f(x)=-240 x^{2}+1,440 x .\) Find the capacity of the gutter if \(x\) is 3 inches. (GRAPH CANNOT COPY)
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