/*! 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 2 Rewrite each quadratic equation ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 2

Rewrite each quadratic equation in general form if necessary. For each equation, identify the values of \(a, b\), and \(c\). a. \(2 x^{2}+3 x-7=0\) b. \(x^{2}+6 x=-11\) (a) c. \(-3 x^{2}-4 x+12=0\) d. \(18-4.9 x^{2}+47 x=0\) (a) e. \(-16 x^{2}+28 x+10=57\)

Short Answer

Expert verified
a) \(a=2\), \(b=3\), \(c=-7\); b) \(a=1\), \(b=6\), \(c=11\); c) \(a=-3\), \(b=-4\), \(c=12\); d) \(a=-4.9\), \(b=47\), \(c=18\); e) \(a=-16\), \(b=28\), \(c=-47\).

Step by step solution

01

Confirm equation a is in general form

The equation given is \(2x^2 + 3x - 7 = 0\). This is already in the general quadratic form \(ax^2 + bx + c = 0\).
02

Identify coefficients for equation a

From the equation \(2x^2 + 3x - 7 = 0\), we identify the coefficients as \(a = 2\), \(b = 3\), and \(c = -7\).
03

Rewrite equation b in general form

The original equation is \(x^2 + 6x = -11\). To rewrite it, add 11 to both sides: \(x^2 + 6x + 11 = 0\).
04

Identify coefficients for equation b

In the equation \(x^2 + 6x + 11 = 0\), the coefficients are \(a = 1\), \(b = 6\), and \(c = 11\).
05

Confirm equation c is in general form

The equation given is \(-3x^2 - 4x + 12 = 0\). It is already in the general form \(ax^2 + bx + c = 0\).
06

Identify coefficients for equation c

In \(-3x^2 - 4x + 12 = 0\), the coefficients are \(a = -3\), \(b = -4\), and \(c = 12\).
07

Rewrite equation d in general form

The original equation is \(18 - 4.9x^2 + 47x = 0\). Rearrange it: \(-4.9x^2 + 47x + 18 = 0\).
08

Identify coefficients for equation d

The equation is \(-4.9x^2 + 47x + 18 = 0\), so \(a = -4.9\), \(b = 47\), and \(c = 18\).
09

Rewrite equation e in general form

The original equation is \(-16x^2 + 28x + 10 = 57\). Subtract 57 from both sides to get: \(-16x^2 + 28x - 47 = 0\).
10

Identify coefficients for equation e

From \(-16x^2 + 28x - 47 = 0\), the coefficients are \(a = -16\), \(b = 28\), and \(c = -47\).

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.

General Form
The general form of a quadratic equation is an essential concept in understanding how to work with quadratics. It is typically written as \( ax^2 + bx + c = 0 \). This form includes three main components:
  • Quadratic term: This is the \( ax^2 \) part, where \( a \) is a coefficient that multiplies the squared variable \( x^2 \).
  • Linear term: This is the \( bx \) part, where \( b \) is a coefficient of the variable \( x \).
  • Constant term: The \( c \) part is just a constant number, not attached to a variable.
Quadratic equations in general form are the foundation for many mathematical operations, such as solving by factoring, completing the square, or using the quadratic formula. Bringing any equation into this form enables a more straightforward application of these methods and helps make comparison and analysis easier.
Coefficients
Coefficients are numbers that multiply variables in an equation, and they play a critical role in defining the behavior and solutions of quadratic equations. In the general form \( ax^2 + bx + c = 0 \), the coefficients \( a \), \( b \), and \( c \) have specific functions:
  • Coefficient \( a \): Determines the direction and curvature of the parabola (upward if \( a > 0 \), and downward if \( a < 0 \)).
  • Coefficient \( b \): Affects the vertex location and shape of the parabola, influencing its horizontal placement on the graph.
  • Constant \( c \): Represents the value of the equation when \( x = 0 \), effectively marking the y-intercept on a graph.
Knowing the coefficients allows us to predict and graph the equation's behavior and further apply methods to find solutions, like roots or zeros of the function.
Rewrite Equations
Rewriting equations into general form is a common algebraic technique that helps simplify the problem-solving process. Sometimes quadratic equations do not initially appear in their general form, so we need to manipulate them algebraically to reformat them. For example, in the equation \( x^2 + 6x = -11 \), we can rewrite it as \( x^2 + 6x + 11 = 0 \) by adding 11 to both sides. This form makes identifying coefficients and solving the equation more straightforward.Rewriting involves:
  • Transposing terms: Move terms from one side to the other to centralize all expressions.
  • Arranging terms: Align the equation in the \( ax^2 + bx + c \) structure.
This process is crucial for ensuring that all quadratic equations follow a consistent format, which aids in systematically applying mathematical methods.
Mathematics Education
Mathematics education involving quadratic equations focuses on developing strong problem-solving skills and conceptual understanding. Learning to identify and work with terms like the general form, coefficients, and rewriting equations enhances a student's ability to tackle more complex math tasks. In an educational setting, studying quadratic equations prepares students for advanced mathematical topics, such as:
  • Calculus: Quadratics are foundational for understanding derivatives and integrals.
  • Physics: Many physical phenomena, like projectile motion, are modeled using quadratic equations.
  • Engineering: Quadratics appear in various engineering calculations, making them vital in practical applications.
Effective mathematics education emphasizes understanding concepts deeply and reinforcing learning through practice, enabling students to apply mathematical knowledge confidently and creatively across disciplines.

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

The Cruisin' Along Company is determining prices for its Caribbean cruise packages. The basic price is \(\$ 2,500\) per person. However, business is slow. To attract corporate clients, the company reduces the price of each ticket by \(\$ 5\) for each person in the group. The larger the group, the less each person pays. a. Define variables and write an equation for the price of a single ticket. (a) b. Write an equation for the total price the company charges for a group package. (a) c. Convert the equation in \(9 \mathrm{~b}\) to vertex form. d. What is the total price of a cruise for a group of 20 people? e. The company accountant reports that the cost of running a cruise is \(\$ 200,000\). Solve the equation $$ x(2500-5 x)=200,000 $$ by completing the square. f. What limitations on group size should the cruise company use in order to make a profit?

Use a symbolic method to solve each equation. Show each solution exactly as a rational or a radical expression. a. \(x^{2}=18\) b. \(x^{2}+3=52\) c. \((x-2)^{2}=25\) d. \(2(x+1)^{2}-4=10\) (a)

A baseball is dropped from the top of a very tall building. The ball's height, in meters, \(t\) seconds after it has been released is \(h(t)=-4.9 t^{2}+147\) a. Find \(h(0)\) and give a real-world meaning for this value. b. Solve \(h(t)=20\) symbolically and graphically. c. Does your answer to 5 b mean the ball is \(20 \mathrm{~m}\) above the ground twice? Explain your reasoning. d. During what interval of time is the ball less than \(20 \mathrm{~m}\) above the ground? (a) e. When does the ball hit the ground? Justify your answer with a graph. (i)

APPLICATION A shopkeeper is redesigning the rectangular sign on her store's rooftop. She wants the largest area possible for the sign. When she considers adding an amount to the width, she subtracts that same amount from the length. Her original sign has width \(4 \mathrm{~m}\) and length \(7 \mathrm{~m}\). a. Complete the table. (hi) \begin{tabular}{|c|c|c|c|c|} \hline Increase \((x)\) \((\mathbf{m})\) & Width \((\mathrm{m})\) & Length \((\mathrm{m})\) & Area \(\left(\mathrm{m}^{2}\right)\) & Perimeter \((\mathrm{m})\) \\\ \hline 0 & 4 & 7 & & \\ \hline \(0.5\) & & & \\ \hline \(1.0\) & & & \\ \hline \(1.5\) & & & \\ \hline \(2.0\) & & & \\ \hline \end{tabular} b. How do the changes in width and length affect the perimeter? c. How do the changes in width and length affect the area? d. Write an equation in factored form for the area \(A\) of the rectangle in terms of \(x\), the amount she adds to the width. e. What are the dimensions of the rectangle with the largest area?

\text { Show a step-by-step symbolic solution of the inequality }-3 x+4>16
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

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.