/*! 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 97 Find the real solutions, if any,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 97

Find the real solutions, if any, of each equation. $$ 3 x^{2}+7 x-20=0 $$

Short Answer

Expert verified
The real solutions are \( x = \frac{5}{3} \) and \( x = -4 \).

Step by step solution

01

Identify the coefficients

In the quadratic equation, identify the coefficients, where the equation is in the form of \(ax^2 + bx + c = 0\). Here, \(a = 3\), \(b = 7\), and \(c = -20\).
02

Apply the quadratic formula

Use the quadratic formula to find the solutions of the equation: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 3\), \(b = 7\), and \(c = -20\) into the formula.
03

Calculate the discriminant

Compute the discriminant, \(D = b^2 - 4ac\). Here, \( D = 7^2 - 4 \cdot 3 \cdot (-20)\), thus, \(D = 49 + 240 = 289\).
04

Calculate the solutions

Since the discriminant is positive, there are real solutions. Use the quadratic formula to find these: \(x = \frac{-7 \pm \sqrt{289}}{6}\), hence, \(x = \frac{-7 \pm 17}{6}\). This gives two solutions: \(x_1 = \frac{-7 + 17}{6} = \frac{10}{6} = \frac{5}{3}\), and \(x_2 = \frac{-7 - 17}{6} = \frac{-24}{6} = -4\).

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 formula
When you're solving quadratic equations, one of the most reliable tools you can use is the quadratic formula. The formula is written as:

  • First, identify the coefficients from your quadratic equation. These coefficients are typically denoted as 'a', 'b', and 'c'.
    In our example, we have the equation:






  • Once you've identified 'a', 'b', and 'c', you can plug them into the quadratic formula:




  • Finally, simplify the results to find the solutions to the quadratic equation.
    For our example, after plugging in the coefficients, we would continue to simplify until we find the values of 'x' that satisfy the equation.





coefficients
Understanding coefficients is crucial when working with quadratic equations. Coefficients are the numerical factors in front of the variables in an equation. In a standard form quadratic equation, which looks like this:

  • 'a' is the coefficient of the term with squared variable, which in our example is 3.

  • 'b' is the coefficient of the term with the single variable, which is 7.

  • 'c' is the constant term, in this case, -20.



Each of these coefficients plays a role in the quadratic formula. Accurately identifying and substituting them into the formula can greatly simplify the problem-solving process. For our given equation:





Remember:

  • Consistency in identifying 'a', 'b', and 'c'.


  • Correctly substituting them into the quadratic formula.

  • Carefully performing the arithmetic operations.
discriminant
The discriminant, represented by 'D', is a key part of the quadratic formula. The discriminant helps determine the nature of the roots of the quadratic equation. The formula for the discriminant is:

  • In our example, where 'b' is 7, 'a' is 3, and 'c' is -20, the discriminant is calculated as follows:



  • The value of the discriminant is 289.



The value of the discriminant tells us:
  • If 'D' is positive, the quadratic equation has two distinct real roots.
  • If 'D' is zero, there is exactly one real root.
  • If 'D' is negative, there are no real roots; instead, there are two complex roots.

In our solved example, the discriminant is 289, a positive value, indicating two distinct real roots. Sure enough, when we solved the equation, we got:





  • -li>;

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

High-Speed Walkways Toronto's Pearson International Airport has a high-speed version of a moving walkway. If Liam walks while riding this moving walkway, he can travel 280 meters in 60 seconds less time than if he stands still on the moving walkway. If Liam walks at a normal rate of 1.5 meters per second, what is the speed of the walkway?

Mixing Water and Antifreeze The cooling system of a certain foreign-made car has a capacity of 15 liters. If the system is filled with a mixture that is \(40 \%\) antifreeze, how much of this mixture should be drained and replaced by pure antifreeze so that the system is filled with a solution that is \(60 \%\) antifreeze?

How much water must be evaporated from 32 ounces of a \(4 \%\) salt solution to make a \(6 \%\) salt solution?

Real Estate A real estate agent agrees to sell an apartment complex according to the following commission schedule: \(\$ 45,000\) plus \(25 \%\) of the selling price in excess of \(\$ 900,000\) Assuming that the complex will sell at some price between \(\$ 900,000\) and \(\$ 1,100,000,\) inclusive, over what range does the agent's commission vary? How does the commission vary as a percent of selling price?

The Social Security Administration determined that an average 30-year-old male in 2018 could expect to live at least 52.2 more years and that an average 30-year-old female in 2018 could expect to live at least 55.8 more years. (a) To what age could an average 30 -year-old male expect to live? Express your answer as an inequality. (b) To what age could an average 30 -year-old female expect to live? Express your answer as an inequality. (c) Who can expect to live longer, a male or a female? By how many years?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

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.