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Chapter 1: Problem 61

Solve each equation using a graphing calculator. [Hint: Begin with the window \([-10,10]\) by \([-10,10]\) or another of your choice (see Useful Hint in Graphing Calculator Terminology following the Preface) and use ZERO, SOLVE, or TRACE and ZOOM IN.] (In Exercises 61 and 62 , round answers to two decimal places.) $$ 2 x^{2}+3 x-6=0 $$

Short Answer

Expert verified
The solutions are approximately \( x \approx 1.35 \) and \( x \approx -2.35 \).

Step by step solution

01

Input the Equation

Enter the given equation into the graphing calculator. Navigate to the function entry screen and type in the equation: \( y = 2x^2 + 3x - 6 \).
02

Set Graphing Window

Set the viewing window to \([-10, 10]\) for both x-axis and y-axis. This ensures the graph displays a good range to identify where it crosses the x-axis.
03

Graph the Function

Graph the equation to see the curve. The function should appear as a parabola because it's a quadratic equation. Look for where the parabola crosses the x-axis.
04

Use Zero Function

Access the ZERO feature on the calculator. This function helps find the x-intercepts, where the graph crosses the x-axis, by selecting points to the left and right of each root.
05

Interpret Graph Results

Using the ZERO feature, find and record the x-values where \( y = 0 \). Ensure the roots are calculated accurately to two decimal places by using the calculator's graphing tool.
06

Verify Solutions

Check the solution by substituting the values back into the equation to ensure they satisfy \( 2x^2 + 3x - 6 = 0 \). If a value eliminates the equation close to zero, it is confirmed as a root.

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Key Concepts

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

Graphing Calculator
A graphing calculator is an indispensable tool in mathematics, especially when dealing with complex equations such as quadratic equations. These calculators are designed to plot mathematical functions, providing visual representations which can aid students in understanding the behavior of these functions.
  • They allow for easy manipulation of equations, enabling students to input and alter parameters to see immediate graphical results.
  • Features such as ZOOM IN and TRACE help to dive into specific areas of interest on the graph, providing a closer look at crucial points like intercepts and bends.
  • The ZERO or SOLVE functions are particularly useful for solving equations as they help find the x-intercepts where the graph meets the x-axis.
For problems involving parabolas, a graphing calculator can simplify the process by visualizing the parabola and finding its roots efficiently.
Function Graphing
Graphing a function involves plotting its values over a certain interval to get a visual picture of its behavior. For quadratic functions, this results in a parabolic shape. Understanding function graphing is key to interpreting equations and predicting their solutions.
  • To graph a function, input its formula into a graphing calculator and choose an appropriate viewing window. For example, setting the axes from [-10, 10] ensures a broad view of the function's behavior around the origin.
  • The calculator then draws the function, showing curves and lines that represent the equation.
  • By following the graph, you can visually identify where the function takes specific values, such as when it crosses the axes.
By graphing, you can also visually verify solutions, finding areas where the function hits zero, identifying maximums or minimums, and observing symmetry.
Quadratic Functions
Quadratic functions, fundamental in algebra, have the form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
  • The graph of a quadratic function is a parabola, symmetric about its vertex. Depending on the sign of \(a\), the parabola opens upwards or downwards.
  • Key features of a quadratic graph include its vertex, axis of symmetry, and intercepts (where it crosses the axes).
  • The x-intercepts, also known as the roots or solutions of the equation, are points where the function equals zero.
To solve quadratic equations, graphing these functions can provide an intuitive understanding of their roots. Use these visual cues to check and understand the factors of the equation and how they contribute to its graphical representation.
X-intercepts
Finding the x-intercepts of a function is crucial as they represent the real solutions to the equation, where the graph crosses the x-axis.
  • For quadratic equations, these intercepts are the solutions or roots, found by setting the function equal to zero, \( y = 0 \).
  • In a graphing calculator, using the ZERO feature helps locate these intercepts precisely. By selecting points left and right around where the graph meets the x-axis, the calculator determines the exact intercept value.
  • Once these points are identified, they can be verified by substituting back into the original equation. If they equal zero, they confirm correct root values.
Understanding x-intercepts is key for solving quadratic equations thoroughly, providing a solid foundation for further analysis in algebra and calculus.

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Most popular questions from this chapter

Write each interval in set notation and graph it on the real line. $$ (-\infty, 2] $$

Write each interval in set notation and graph it on the real line. $$ [0,6) $$

For each pair of functions \(f(x)\) and \(g(x)\), find and fully simplify a. \(f(g(x))\) and b. \(g(f(x))\) $$ f(x)=x^{3}+1 ; \quad g(x)=\sqrt[3]{x-1} $$

How will the graph of \(y=-(x-4)^{2}+8\) differ from the graph of \(y=-x^{2} ?\) Check by graphing both functions together.

Identify each function as a polynomial, a rational function, an exponential function, a piecewise linear function, or none of these. (Do not graph them; just identify their types.) $$ f(x)=5 $$
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