/*! 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 116 Find the \(x\) -intercept \((s)\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 116

Find the \(x\) -intercept \((s)\) of the graph of the function. $$g(x)=x^{2}-x-6[3.2]$$

Short Answer

Expert verified
The x-intercepts are 3 and -2.

Step by step solution

01

- Set the function equal to zero

To find the x-intercepts, set the function equal to zero because at the x-intercepts, the value of the function is zero. So, solve the equation: g(x) = 0 ::= x^2 - x - 6 = 0
02

- Factor the quadratic equation

Factor the quadratic equation on the right side. Look for two numbers that multiply to -6 and add up to -1. The equation factors to: (x - 3)(x + 2) = 0
03

- Solve for x

Set each factor equal to zero and solve for x. This gives: x - 3 = 0 x = 3 x + 2 = 0 x = -2

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
A quadratic equation is a type of polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x represents the variable. The highest exponent in a quadratic equation is 2, making it a second-degree polynomial.

Quadratic equations can often have two solutions because the graph of a quadratic function is a parabola, which can intersect the x-axis at zero, one, or two points. The points where a parabola crosses the x-axis are known as the x-intercepts.

Here are some common forms of quadratic equations:
  • Standard Form: ax^2 + bx + c = 0
  • Factored Form: a(x - p)(x - q) = 0, where p and q are the solutions or roots.
  • Vertex Form: a(x - h)^2 + k = 0, where (h, k) is the vertex of the parabola.
factoring
Factoring is a method used to break down complex expressions into simpler ones that can be multiplied together to get the original expression. In the context of quadratic equations, factoring is used to find the solutions or roots.

For the quadratic equation g(x) = x^2 - x - 6, we look for two numbers that multiply to -6 (the constant term) and add up to -1 (the coefficient of the x term). These numbers are -3 and 2. Therefore, the quadratic equation can be factored into:

(x - 3)(x + 2) = 0

Factoring turns the quadratic equation into a product of two binomials, each set to zero.
  • Factored form of x^2 - x - 6: (x - 3)(x + 2) = 0
This decomposition makes it easier to solve for the x-values where the expression is zero.
solving equations
Solving equations is the process of finding the values of variables that satisfy the given equation. In our specific problem, we need to solve for the values of x that make the function g(x) = 0.

Once g(x) = x^2 - x - 6 is factored, we have the equation in the form:

(x - 3)(x + 2) = 0

To find the solutions, we set each factor equal to zero:
  • x - 3 = 0 --> x = 3
  • x + 2 = 0 --> x = -2
This means the x-intercepts of the graph of g(x) are x = 3 and x = -2. These are the points where the parabola crosses the x-axis.

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

Make a hand-drawn graph of the function. Then check your work using a graphing calculator. $$h(x)=\ln x$$

Given that \(\sec \beta=1.5304,\) find \(\sin \left(90^{\circ}-\beta\right)\)

Find the reference angle and the exact function value if they exist. $$\cos 0^{\circ}$$

The transformation techniques that we learned in this section for graphing the sine and cosine functions can also be applied to the other trigonometric functions. Sketch a graph of each of the following. Then check your work using a graphing calculator. $$y=\cot (2 x)$$

Given that \(\sin 38.7^{\circ} \approx 0.6252, \cos 38.7^{\circ} \approx 0.7804\) and \(\tan 38.7^{\circ} \approx 0.8012,\) find the six function values of \(51.3^{\circ}\).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Decision Maths

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.