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Chapter 11: Problem 58

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. See Section 11.1. $$ x^{2}-10 x $$

Short Answer

Expert verified
Add 25 to the binomial to form a perfect square trinomial.

Step by step solution

01

Understand the Form of a Perfect Square Trinomial

A perfect square trinomial is of the form \((a - b)^2 = a^2 - 2ab + b^2\). Our task is to identify the value of \(b^2\) to make \(x^2 - 10x\) a perfect square trinomial.
02

Identify the Coefficient to Match with \(-2ab\)

In our binomial, \(-10x\) is equivalent to \(-2ab\). Therefore, the value of \(2ab = 10\). Since the formula is \(a^2 - 2ab + b^2\), here \(a = x\).
03

Solve for \(b\)

We have the equation \(-2x \, b = -10\) or \(2xb = 10\). Solve for \(b\) to get \(b = \frac{10}{2}\) which simplifies to \(b = 5\).
04

Calculate \(b^2\)

Now that we know \(b = 5\), calculate \(b^2\). So, \(b^2 = 5^2 = 25\).
05

Form the Perfect Square Trinomial

Add \(b^2 = 25\) to the original binomial \(x^2 - 10x\) to form the trinomial \(x^2 - 10x + 25\) which is a perfect square, seen as \((x - 5)^2\).

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Key Concepts

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

Binomial
A binomial is an algebraic expression that contains exactly two terms. In the exercise above, the binomial we start with is \(x^2 - 10x\). Understanding binomials is important in manipulating algebraic expressions, especially when transforming them into other forms like perfect square trinomials.
The key to working with binomials lies in:
  • Recognizing the two distinct terms involved. Here we have \(x^2\) and \(-10x\).
  • Understanding how these terms interact with each other, particularly when trying to form other kinds of expressions.
Every term in a binomial serves a purpose in the broader algebraic manipulations, making it essential to identify coefficients and variable parts correctly.
Solving for Constants
Solving for constants involves finding specific values that satisfy a condition within an equation or expression. In transforming the binomial \(x^2 - 10x\) into a perfect square trinomial, we need to solve for the constant \(b\).
To solve for this constant, consider:
  • The relationship described by the expression \(-2ab = -10\).
  • Set \(a = x\) and isolate \(b\) in the equation \(2x \cdot b = 10\).
  • This isolation leads to finding \(b = \frac{10}{2} = 5\).
This calculated constant \(b = 5\) is crucial as it helps us identify \(b^2 = 25\) for completing the square in the original binomial expression.
Trinomial Equations
Trinomial equations are algebraic expressions containing three terms. These can often be rewritten or factored into different forms for solving or simplification. In the current task, we transform a binomial into a trinomial that can be expressed as a perfect square.
A perfect square trinomial takes the form \(a^2 - 2ab + b^2\). By adding the constant term \(b^2\) to our binomial, we achieve this form. Doing so not only completes the square but also allows us to express the trinomial effectively as
  • \((x - 5)^2 = x^2 - 10x + 25\)
  • This demonstrates the inherent symmetry and structure in trinomials.
Mastering trinomial equations aids in solving quadratic equations and other complex algebraic problems efficiently.
Algebraic Expressions
Algebraic expressions combine numbers, variables, and operation symbols to denote a quantity. Learning to manipulate these expressions is foundational in algebra. When dealing with algebraic expressions:
  • Recognize the variables and constants involved.
  • Use operations like addition, subtraction, and multiplication appropriately.
  • Apply transformations, such as forming perfect squares, where applicable.
In our original exercise, identifying and transforming the algebraic expression from \(x^2 - 10x\) to \(x^2 - 10x + 25\) showcases this manipulation, resulting in a simpler, more solvable form \((x - 5)^2\). Understanding these principles enhances your ability to solve and interpret algebraic problems effectively.

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Most popular questions from this chapter

Find the maximum or minimum value of each function. Approximate to two decimal places. \(f(x)=2.3 x^{2}-6.1 x+3.2\)

Sketch the graph of each function. See Section 11.5 $$ f(x)=(x-3)^{2} $$

Use the quadratic formula to solve each quadratic equation. $$ 7 x^{2}+\sqrt{7} x-2=0 $$

Use the quadratic formula to solve each quadratic equation. $$ x^{2}+\sqrt{2} x+1=0 $$

Solve by completing the square. See Section 11.1. $$ z^{2}+10 z-1=0 $$
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