Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 35

A parabola described by the equation \(y=x^2-6 x+c\) is intersected exactly once in the \(x y\)-plane by the equation \(y=-1\). What is the value of \(c\) ?

Short Answer

Expert verified
The value of c is 8.

Step by step solution

01

Set the two equations equal to each other

Since we are looking for the intersection points where the two functions have the same y value, we can set the equations equal: \(x^2 - 6x + c = -1\)
02

Rearrange and simplify the equation

Rearrange the equation for x in terms of c: \(x^2 - 6x + (c + 1) = 0\)
03

Determine the discriminant

In a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant (\(D\)) is given by the formula \(D = b^2 - 4ac\). In our case, \(a = 1\), \(b = -6\) and \(c = (c + 1)\). Compute the discriminant, D: \(D = (-6)^2 - 4(1)(c + 1)\)
04

Set the discriminant equal to 0

Since we want the parabola to intersect exactly once, this means that there should be only one real solution. This happens when the discriminant is equal to 0: \((-6)^2 - 4(1)(c + 1) = 0\)
05

Solve for c

Now solve the equation for c: \(36 - 4(c + 1) = 0\) \(36 - 4c - 4 = 0\) \(32 - 4c = 0\) \(4c = 32\) \(c = 8\) So, the value of c is 8.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

For the function defined as \(g(x)=\frac{x-2}{3}\), what is the value of \(g(-4)\) ? A) \(-2\) B) \(-\frac{2}{3}\) C) \(\frac{2}{3}\) D) 2

The writer wants to include a detail that will clarify the phrase "dayto-day operations." Which of the following would best fulfill this goal? A) NO CHANGE B) and some that are more long-term, C) often repetitive tasks, D) not the political decisionmaking,

A) NO CHANGE B) Brown's name became a rallying cry for both sides as the nation moved toward Civil War. C) Brown's cry was the rally that both sides named as the Civil War moved in on the nation. D) Both sides used Brown's name as the nation's Civil War was moving toward it.

A right square pyramid has a volume of 324 cubic centimeters and a height of 12 centimeters. What is the length of one side of the base, in centimeters? A) \(18 \sqrt{2}\) B) 18 C) 9 D) \(3 \sqrt{3}\)

John is buying some burgers and fries for his friends. Burgers cost $$\$ 4.30$$ each and fries cost $$\$ 3.10$$ per order. There is no tax on the food. If he has $$\$ 50$$ and buys 6 orders of fries, what is the maximum number of burgers he can buy?
See all solutions

Recommended explanations on English Textbooks

Single Paragraph Essay

Read Explanation

Language Analysis

Read Explanation

Listening and Speaking

Read Explanation

Argumentative Essay

Read Explanation

Essay Prompts

Read Explanation

English Grammar Summary

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.

Sign-up for free