/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 93 Find \(b\) such that each trinom... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 93

Find \(b\) such that each trinomial is a square. $$ x^{2}+b x+81 $$

Short Answer

Expert verified
b = 18 or b = -18

Step by step solution

01

- Identify the form of a perfect square trinomial

A perfect square trinomial takes the form: \[ (x + a)^2 = x^2 + 2ax + a^2 \]We need to match this form to the given trinomial.
02

- Compare the given trinomial with the perfect square form

The given trinomial is: \[ x^2 + bx + 81 \]Notice that comparing our given trinomial to the perfect square form:\[ x^2 + 2ax + a^2 \]we can see that the term 81 corresponds to \(a^2\), i.e., \[ a^2 = 81 \]
03

- Solve for 'a'

To find 'a', take the square root of both sides of the equation: \[ a = \pm \sqrt{81} = \pm 9 \]
04

- Find 'b' using the value of 'a'

From the perfect square trinomial form, we know that \[ b = 2a \]Now substitute the values of 'a' to find 'b':For \( a = 9 \):\[ b = 2(9) = 18 \]For \( a = -9 \):\[ b = 2(-9) = -18 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Trinomial
A trinomial is an algebraic expression composed of three terms. It's a type of polynomial, and each term is connected by addition or subtraction. For instance, in the trinomial \(x^2 + bx + 81\), we have three distinct parts: \(x^2\), \(bx\), and 81. Trinomials are common in algebra and are often used in problems involving quadratic equations. To solve these problems, you need to understand how to handle each term effectively. In many cases, you'll seek to rewrite or manipulate trinomials to make them easier to solve or factor.
Quadratic Equations
A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\) where \(a, b,\text{ and }c\) are constants, and \(a eq 0\). Quadratic equations describe parabolic graphs and can be solved using various methods, such as factoring, completing the square, or the quadratic formula. In our exercise, the given trinomial \(x^2 + bx + 81\) is a quadratic expression, and we need to find the value of \(b\) so that it fits the format of a perfect square trinomial. Perfect square trinomials can be factored into the square of a binomial, which helps simplify the equation.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They are the building blocks of algebra. Understanding how to manipulate and simplify these expressions is crucial for solving complex problems. Let's look closer at the perfect square trinomial form, \((x + a)^2 = x^2 + 2ax + a^2\). When given \(x^2 + bx + 81\), we compare it to the standard form and identify that 81 is \(a^2\). By taking the square root of 81, we find \(a\). Using \(a\), we can then derive \(b = 2a\). This process is a classic example of working with algebraic expressions to find specific values that make the equation true.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Two pipes are connected to the same tank. Working together, they can fill the tank in 4 hr. The larger pipe, working alone, can fill the tank in 6 hr less time than it would take the smaller one. How long would the smaller one take, working alone, to fill the tank?

Height of a Thrown Object. The function $$S(t)=-16 t^{2}+32 t+1920$$ gives the height \(S\), in feet, of an object thrown from a cliff that is 1920 ft high. Here \(t\) is the time, in seconds, that the object is in the air. a) For what times does the height exceed 1920 ft ? b) For what times is the height less than 640 ft?

What would be better: to receive \(3 \%\) interest every 6 months, or to receive \(6 \%\) interest every 12 months? Why?

Write the equation of the parabola that has the shape of \(f(x)=2 x^{2}\) or \(g(x)=-2 x^{2}\) and has a maximum value or a minimum value at the specified point. $$ \text { Minimum: }(-4,0) $$

Rational Inequalities Solve. For \(F(x)=\frac{1}{x-3},\) find all \(x\) -values for which \(F(x) \leq 2\).
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.