/*! 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 19 One of the roots is given. Find ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 19

One of the roots is given. Find the other root. \(x^{2}-8 x+c=0 ;-3\)

Short Answer

Expert verified
The other root is 11.

Step by step solution

01

Understand the Problem

We are given a quadratic equation \(x^{2} - 8x + c = 0\), and one of the roots is \(-3\). We need to find the other root of this equation.
02

Use the Root to Find c

Since \(-3\) is a root of the equation, substitute \(x = -3\) into the equation to find \(c\). This gives: \((-3)^{2} - 8(-3) + c = 0\), simplifying to \(9 + 24 + c = 0\). Thus, \(c = -33\).
03

Write the Complete Equation

With \(c = -33\), the equation becomes \(x^{2} - 8x - 33 = 0\).
04

Use Vieta's Formulas

According to Vieta's formulas, the sum of the roots \(r_1 + r_2\) of a quadratic equation \(x^2 + bx + c = 0\) is \(-b\) (which is 8 in this case), and the product \(r_1 \cdot r_2\) is \(c\) (which is -33). With \(r_1 = -3\), we solve \(-3 + r_2 = 8\), so \(r_2 = 11\).
05

Verify the Solution

Check that the roots \(-3\) and \(11\) satisfy the original equation \(x^2 - 8x - 33 = 0\). The product of the roots is \((-3) \times 11 = -33\), which matches the constant term, and the sum is \(-3 + 11 = 8\), which matches the negative coefficient of \(x\). Thus, both calculations confirm the solution.

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.

Vieta's Formulas
Understanding Vieta's formulas is a great way to find relationships between the roots and coefficients of quadratic equations.

For any quadratic equation framed as \(x^2 + bx + c = 0\), Vieta's formulas help us determine two things:
  • The sum of the roots \(r_1 + r_2\) is equal to \(-b\).
  • The product of the roots \(r_1 \cdot r_2\) is equal to \(c\).
Applying this to our equation \(x^2 - 8x - 33 = 0\), you see that the sum of the roots must be 8 (from \(-(-8)\)) and their product must be \(-33\). With one root known as \(-3\), you can use these relationships to efficiently find the other root. This is a useful tool for solving many quadratic equation problems quickly.
Finding Roots
Finding the roots of a quadratic equation like \(x^2 - 8x - 33 = 0\) involves determining the values of \(x\) that satisfy the equation.

With one root given, such as \(-3\), use it to identify constants and find the other root by using algebraic methods or Vieta's formulas.
  • Start with the sum of the roots formulation \(-3 + r_2 = 8\).
  • This means \(r_2 = 11\).
These roots \(-3\) and \(11\) fit perfectly due to their sum \(8\) and product \(-33\). Remember, each root represents a solution where the expression \(x^2 - 8x - 33 = 0\) equals zero.
Substituting Values
Substituting known values allows us to solve for unknowns or verify solutions in a quadratic equation.

When given that \(-3\) is a root of \(x^2 - 8x + c = 0\), plug \(x = -3\) into the equation:
  • \((-3)^2 - 8(-3) + c = 0\).
  • Simplifying yields \(9 + 24 + c = 0\), so \(c = -33\).
Substituting values is crucial for manipulating equations to find other missing parts. Once \(c\) is found, the equation turns into \(x^2 - 8x - 33 = 0\). By substituting \(-3\) and \(11\) back into this modified equation, you can confirm they satisfy it, affirming these values as the correct roots.

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

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{y=2 x^{2}+3 x+4} \\ {y=11 x+6}\end{array} $$

In \(19-28 :\) a. Find \(\mathrm{f}(a)\) for each given function. b. Is \(a\) a root of the function? $$ \mathrm{f}(x)=-x^{3}+x-24 \text { and } a=-3 $$

In \(44-51 :\) a. Graph the given inequality. b. Determine if the given point is in the solution set. $$ -4(x+2)^{2}-5 \leq y ;\left(1, \frac{2}{3}\right) $$

One of the roots is given. Find the other root. \(m^{2}-4 m+n=0 ; 3\)

In \(9-17,\) graph each system and determine the common solution from the graph. $$ \begin{array}{l}{-x^{2}+4 x-y-2=0} \\ {x+y=4}\end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics 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.