/*! 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 23 Use the quadratic formula to sol... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 23

Use the quadratic formula to solve each equation. $$ 4 k^{2}=5 k+2 $$

Short Answer

Expert verified
The solutions are \(k = \frac{5 + \sqrt{57}}{8}\) and \(k = \frac{5 - \sqrt{57}}{8}\).

Step by step solution

01

- Rewrite the equation in standard form

The quadratic equation must be in the form of \(ax^2 + bx + c = 0\). Start by moving all terms to one side so that the equation equals zero.$$4k^2 - 5k - 2 = 0$$
02

- Identify coefficients

In the standard form equation \(ax^2 + bx + c = 0\), identify the values of \(a\), \(b\), and \(c\). Here, \(a = 4\), \(b = -5\), and \(c = -2\).
03

- Write the quadratic formula

The quadratic formula is given by: \[k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
04

- Substitute the values into the quadratic formula

Insert the values of \(a\), \(b\), and \(c\) into the quadratic formula:\[k = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(4)(-2)}}{2(4)}\]
05

- Simplify under the square root

First, calculate the expression under the square root (the discriminant):\[(-5)^2 - 4(4)(-2) = 25 + 32 = 57\]
06

- Simplify the entire expression

Now, simplify the expression:\[k = \frac{5 \pm \sqrt{57}}{8}\]
07

- Solve for the values of k

Considering both the positive and negative square roots, the solutions to the quadratic equation are:\[k = \frac{5 + \sqrt{57}}{8}\] and \[k = \frac{5 - \sqrt{57}}{8}\]

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.

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.

Solving Quadratic Equations
Quadratic equations form a fundamental part of algebra. They look like this: \(ax^2 + bx + c = 0\).
The goal is to find the values of x (or any variable used) that make the equation true.
We usually have three main methods to solve them:
  • Factoring
  • Completing the Square
  • Quadratic Formula
For this exercise, we use the Quadratic Formula since it's very efficient and works on all quadratic equations. Understanding each step is key to mastering this formula.
Discriminant
The discriminant is a key part of the quadratic formula. Found under the square root, it's represented by \(b^2 - 4ac\).
This value helps determine the nature of the solutions to the quadratic equation:
  • If the discriminant is positive, you get two real and distinct solutions.
  • If it's zero, you get exactly one real solution (a repeated root).
  • If it's negative, there are no real solutions. Instead, you get two complex solutions.
So for our exercise, the discriminant \[(-5)^2 - 4(4)(-2) = 57\]
Since 57 is positive, we know the quadratic equation has two distinct real solutions.
Standard Form of a Quadratic Equation
A quadratic equation must be in standard form before using the quadratic formula.
The standard form is written as \[ax^2 + bx + c = 0\].
In the given exercise, the initial equation \[4k^2=5k+2\] needs rearranging to look like standard form.
Here鈥檚 how we do it:
  • Move all terms to one side so the right-hand side is zero.
  • This gives us: \[4k^2 - 5k - 2 = 0\]
Now, it鈥檚 much easier to identify the coefficients \[a = 4, b = -5, c = -2\].
Once in standard form, we can systematically apply the quadratic formula.

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

The 鈥淪huffle鈥� button on Tamika鈥檚 CD player plays the songs in a random order. Tamika puts a four-song CD into the player and presses 鈥淪huffle.鈥� a. How many ways can the four songs be ordered? b. What is the probability that Song 1 will be played first? c. What is the probability that Song 1 will not be played first? d. Songs 2 and 3 are Tamika鈥檚 favorites. What is the probability that one of these two songs will be played first? e. What is the probability that Songs 2 and 3 will be the first two songs played (in either order)?

Kai wants to create a fair dice game for two people, using divisibility by \(3 .\) He decided on these rules: \(\cdot\) Each player rolls one die. \(\cdot\) The players find the sum of the two numbers. \(\cdot\) Player 1 scores if the sum is divisible by \(3 .\) Player 2 scores if the sum is not divisible by \(3 .\) Kai isn't sure how many points each player should score each time. a. Make a table showing the probability of each sum being rolled. b. What is the probability that Player 1 scores on a given roll? c. Kai decided to give 2 points to Player 2 when the sum isn't divisible by \(3 .\) For the game to be fair, how many points should Player 1 score each time the sum is divisible by 3\(?\)

Suppose the \(3-o f-7\) lottery game was modified so that after each number was selected, that number was placed back into the group before the next number was selected. In this way, a number could be repeated, meaning triples such as \(1-2-2\) and \(3-3-3\) would be possible. a. How many possible pairs are there for this modified game, assuming that order does matter? Explain. b. Since order really doesn't matter in this game, \(1-1-2,1-2-1,\) and \(2-1-1\) are all the same triple. So there are only 84 possible different triples. Are all of these different triples equally likely? Explain. c. If you choose one number triple for this modified game, what is the probability you will win. (Hint: There are three cases to consider.)

Find the \(x\) -intercepts for the graph of each equation. $$ (x-5)^{2}-49=0 $$

Factor each expression. $$ -3 x^{2}+3 x+18 $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Pure 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.