/*! 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 132 Factor the polynomial. $$16 x^... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 132

Factor the polynomial. $$16 x^{2}-54 x-7$$

Short Answer

Expert verified
Upon finding the roots using the quadratic formula from step 2, the quadratic can be factored as in step 3. However, the roots are dependent on the solutions acquired from the quadratic formula as there isn't a straight forward method to factor the equation.

Step by step solution

01

Identify the quadratic

First, it can be seen that the equation given is a quadratic equation in the form of \(ax^{2} + bx + c\), where \(a = 16\), \(b = -54\), and \(c = -7\)
02

Apply the quadratic formula

For any quadratic expression in the form \(ax^{2} + bx + c\), the roots can be found using the quadratic formula, which is \(-b \pm \sqrt{b^{2}-4ac} / 2a\). When we plug \(a = 16\), \(b = -54\), and \(c = -7\) into this equation, we get the roots.
03

Factorize the quadratic expression

Knowing the roots of the corresponding quadratic equation, we can express the original quadratic expression in factored form. If we denote the roots by \(r_{1}\) and \(r_{2}\), then the factored version of the quadratic expression will be \(a(x-r_{1})(x-r_{2})\), to produce the equivalent form of the original given expression.

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.

Quadratic Equations
Quadratic equations are quite common in algebra, and they usually show up in the standard form: \( ax^2 + bx + c = 0 \). Why are they called 'quadratic'? Because the term "quadratic" comes from "quad," which means square. Quadratics involve the square of a variable, typically \( x^2 \). They hold significance because they can model real-world situations such as projectile motion or calculating areas.
  • They always result in a parabolic graph which could be opening upwards or downwards.
  • The solution or "roots" of a quadratic are the \( x \) values for which the expression equals zero.
  • Quadratics can have either zero, one, or two real solutions.
Understanding the components \( a, b, \) and \( c \) helps in solving these equations. These coefficients guide us on how "wide" or "narrow" the parabola is, and how it is situated on the graph.
Quadratic Formula
The quadratic formula is a trustworthy tool for finding the roots of any quadratic equation. Derived from the process of completing the square, it is often a "go-to" method when factoring seems challenging. Given
\[ ax^2 + bx + c = 0 \]
The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula allows you to calculate the roots by simply plugging in the values from the given quadratic equation.
  • The term under the square root, \( b^2 - 4ac \), is known as the discriminant.
  • The discriminant determines the nature of the roots:
    - If \( b^2 - 4ac > 0 \), there are two distinct real roots.
    - If \( b^2 - 4ac = 0 \), there is exactly one real root.
    - If \( b^2 - 4ac < 0 \), the roots are complex and not real numbers.
Using the quadratic formula might seem a little complicated at first, but it becomes straightforward with practice. It provides a reliable solution pathway when other methods are either difficult or infeasible.
Factoring Techniques
Factoring is a method used to express a polynomial as a product of its simpler polynomials, or "factors". This technique is especially useful because once a quadratic is factored, it's easy to find the roots by setting each factor equal to zero.
There are several techniques for factoring quadratics:
  • **Trial and Error:** This intuitive method involves guessing and checking possible factors.
  • **Using the Quadratic Formula:** Once the roots are known, the quadratic can be expressed in factored form \( a(x - r_1)(x - r_2) \), where \( r_1 \) and \( r_2 \) are the roots.
  • **Difference of Squares:** This involves recognizing terms that can be expressed as \( (x-a)(x+a) \).
  • **Factoring by Grouping:** Used when a quadratic expression does not have a leading coefficient of 1. Group terms to find common factors and simplify the expression.
Factoring requires understanding and practice, but it becomes a powerful technique especially when graphing or solving practical problems. Once mastered, it paves the path to simpler solution methods for solving quadratic equations.

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

Use a graphing utility to approximate the point of intersection of the graphs. Round your result to three decimal places. $$\begin{array}{l}y_{1}=4 \\\y_{2}=3^{x+1}-2\end{array}$$

Divide using synthetic division. $$\left(x^{4}-3 x+1\right) \div(x+5)$$

(a) complete the table to find an interval containing the solution of the equation, (b) use a graphing utility to graph both sides of the equation to estimate the solution, and (c) solve the equation algebraically. Round your results to three decimal places. $$6 \log _{3}(0.5 x)=11$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & 12 & 13 & 14 & 15 & 16 \\\\\hline 6 \log _{3}(0.5 x) & & & & & \\\\\hline\end{array}$$

Find the time required for a \(\$ 1000\) investment to (a) double at interest rate \(r,\) compounded continuously, and (b) triple at interest rate \(r\), compounded continuously. Round your results to two decimal places. $$r=6 \%$$

Use the regression feature of a graphing utility to find an exponential model \(y=a b^{x}\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(0,4.0),(2,6.9),(4,18.0),(6,32.3),(8,59.1),(10,118.5)$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.