/*! 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 51 Find all integers \(b\) such tha... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 51

Find all integers \(b\) such that the trinomial can be factored. $$ 2 x^{2}+b x-6 $$

Short Answer

Expert verified
The integers \(b\) for which the trinomial \(2x^{2}+bx-6\) can be factored are \(-10, -2, 2, 10\).

Step by step solution

01

Determining the discriminant

First, we need to determine the discriminant (\(b^2 - 4ac\)). Since in this case, \(a = 2\) and \(c = -6\), plug the values into the discriminant. Thus, the discriminant is \(b^{2} - 4*2*-6.\)
02

Setting the discriminant as a perfect square

In order for the trinomial to be factored, the discriminant should be a perfect square. So, set \(b^{2} - 4*2*-6\) equal to \(s^{2}\), where \(s\) is any integer and solve for \(b\). This results into two equations: \(b^{2} - 24 = s^{2}\) and \(b = \sqrt{s^{2} + 24}\) or \(b = - \sqrt{s^{2} + 24}\).
03

Finding all integer values of b

The value of \(b\) will be determined by all integer values of \(s\) that make \(s^{2} + 24\) a perfect square. By testing various integers, find that \(b\) can be \(-10, -2, 2, 10\).

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.

Discriminant in Algebra
The discriminant is a crucial part of solving quadratic expressions. It's represented by the formula \(b^2 - 4ac\). The values of \(a\), \(b\), and \(c\) are the coefficients from the quadratic expression \(ax^2 + bx + c\). The discriminant helps to determine the nature of the roots of the expression. When:
  • The discriminant is positive, there are two distinct real roots.
  • Zero, there is exactly one real root.
  • Negative, there are no real roots.
In our exercise, plugging values \(a = 2\) and \(c = -6\) into the discriminant formula shows that the expression's nature depends heavily on the variable \(b\). Understanding discriminants makes it easier to predict when and how a quadratic can be factored.
Perfect Squares
A perfect square is a number that is the square of an integer. For example, 4 is a perfect square because it equals \(2^2\). Recognizing perfect squares is key when solving quadratic expressions, especially when using the discriminant.

In the context of factoring a trinomial, the discriminant should be a perfect square. This ensures the expression can be simplified into two binomials. By setting \(b^2 - 24 = s^2\), where \(s\) is an integer, we find when the discriminant is a perfect square.

This process is essential for identifying integers \(b\) that make the trinomial factored smoothly.
Integer Solutions
When solving quadratics, integer solutions are values that make the expression whole numbers easily. For our exercise, solving for \(b\) involved ensuring the discriminant became a perfect square integer: \(b^2 - 24 = s^2\).

Testing integers for \(s\) helped reveal the values of \(b\) that work. The goal was for \(b\) to keep \(s^2 + 24\) as a perfect square. Trials with integers like \(s = 8, 4, -4, -8\) yield integer solutions for \(b\) such as -10, -2, 2, and 10.
  • This step involves testing and calculating, ensuring all possibilities are checked.
Finding integer solutions in this way gives a clear path to factor trinomials.
Quadratic Expressions
Quadratic expressions are polynomials of degree two, presented as \(ax^2 + bx + c\). These expressions are foundational in algebra and appear frequently in both academic and real-world contexts.

Factoring quadratic expressions simplifies them into products of binomials. For example, breaking down \(2x^2 + bx - 6\).
  • The first step is finding values that fit into known algebraic identities.
  • A common method to factor these is ensuring the discriminant is a perfect square.
Pay attention to each part: the coefficient of \(x^2\), the linear term \(b\), and the constant term. Understanding how these interact helps uncover hidden patterns and solutions.

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

Factor the polynomial completely. (Note: Some of the polynomials may be prime.) $$ (x-3)^{2}-100 $$

Find two consecutive positive even integers whose product is 440 .

Determine whether the statement is true or false. Justify your answer. $$ x^{2}+4=(x+2)^{2} $$

Find two consecutive positive integers whose product is 240 .

The length of a rectangular painting is 3 inches greater than its width (see figure). The area of the painting is 108 square inches. Find the dimensions of the painting.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.