/*! 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 10 Solve each equation by graphing.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 10

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. $$ 4 x^{2}-7 x-15=0 $$

Short Answer

Expert verified
Roots are between \(-1\text{ and }0\) and \(3\text{ and }4\).

Step by step solution

01

Identify the function

The equation given is \( 4x^2 - 7x - 15 = 0 \). This is a quadratic equation. To solve it by graphing, we need to consider it as a function: \( y = 4x^2 - 7x - 15 \).
02

Create a table of values

Choose a range of values for \( x \), both negative and positive, and calculate corresponding \( y \) values using \( y = 4x^2 - 7x - 15 \). For example, if \( x = -3, -2, -1, 0, 1, 2, 3 \), compute each \( y \).
03

Plot the points on a graph

Using the calculated \( (x, y) \) points, plot them on a coordinate plane. This will form the parabola of the function \( y = 4x^2 - 7x - 15 \).
04

Identify the x-intercepts

The x-intercepts are the points where the curve crosses the x-axis, i.e., where \( y = 0 \). These are the solutions to the equation \( 4x^2 - 7x - 15 = 0 \).
05

Estimate the roots

From the graph, observe the x-coordinates where the parabola crosses the x-axis. These will not provide exact values but will give the approximate intervals. For instance, the graph suggests roots between \( x = -1 ext{ and } 0 \) and \( x = 3 ext{ and } 4 \).
06

State the consecutive integers

Since the exact roots could not be easily found, state the integers between which the roots are located based on the x-intercepts: between \(-1 ext{ and }0\) and \(3 ext{ and }4\).

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.

Graphing Quadratic Functions
When solving quadratic equations, graphing the related quadratic function is a useful visual method. A quadratic function typically appears as a parabola on a coordinate plane, due to its characteristic equation form, \( ax^2 + bx + c = 0 \). To graph a quadratic function, follow these steps:
  • Identify the quadratic equation, such as \( y = 4x^2 - 7x - 15 \).
  • Create a table of values by selecting a range of \( x \) values to compute the corresponding \( y \) values.
  • Plot these points on a graph. The shape that emerges is a parabola.
Ensure to include both negative and positive \( x \) values to get a comprehensive view of the parabola's shape.
X-Intercepts
Finding the x-intercepts of a quadratic function graph is crucial for solving the equation. The x-intercepts are points on the graph where \( y = 0 \). In essence, these points represent the solutions or roots of the quadratic equation.
  • After plotting the parabola, look for the points where it crosses the x-axis.
  • These x-values are your estimated solutions to \( 4x^2 - 7x - 15 = 0 \).
Precise x-intercepts can be tricky to determine through graphing alone, often providing an interval rather than exact values.
Parabolas
Parabolas are U-shaped curves that graphically represent quadratic functions. The main features of a parabola include:
  • Vertex: The lowest or highest point, depending on the opening direction.
  • Axis of symmetry: A vertical line that divides the parabola into two mirror-image halves.
  • Direction of opening: Determined by the sign of the coefficient \( a \). If \( a > 0 \), it opens upwards; if \( a < 0 \), downwards.
Understanding the components of a parabola helps in predicting its shape and position on the graph.
Consecutive Integers
In some cases, the x-intercepts of a parabola as observed on a graph are not clearly defined. This often occurs when the roots are not whole numbers. Here, stating consecutive integers provides an approximate solution.
  • If the parabola crosses between \( x = -1 \) and \( x = 0 \), say the root is between these integers.
  • The same method applies if it crosses between \( x = 3 \) and \( x = 4 \).
Specifying consecutive integers helps to give a clearer idea of where the exact roots might be within an interval.

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

Solve each equation by using the method of your choice. Find exact solutions. \(2 x^{2}-12 x+7=5\)

If \(2 x^{2}-5 x-9=0,\) then \(x\) could be approximately equal to which of the following? A. \(-1.12\) B. 1.54 C. 2.63 D. 3.71

Solve each equation by using the method of your choice. Find exact solutions. \(-4(x+3)^{2}=28\)

Solve each equation by using the Square Root Property. \(x^{2}+18 x+81=25\)

For Exercises \(43-45,\) use the following information.The girls' softball team is sponsoring a fund-raising trip to see a professional baseball game. They charter a \(60-\) passenger bus for \(\$ 525 .\) In order to make a profit, they will charge \(\$ 15\) per person if all seats on the bus are sold, but for each empty seat, they will increase the price by \(\$ 1.50\) per person. What is the minimum number of passengers needed in order for the softball team not to lose money?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

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.