/*! 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 70 $$ \text { For Problems 51-80,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 70

$$ \text { For Problems 51-80, solve each equation. (Objective 2) } $$ $$ 5 x(5 x+2)=8 $$

Short Answer

Expert verified
The solutions are \(x = \frac{2}{5}\) and \(x = -\frac{4}{5}\).

Step by step solution

01

Expand the Equation

Start by expanding the left side of the equation. Multiply the terms in the expression \(5x(5x + 2)\). This results in \(5x \cdot 5x + 5x \cdot 2\). Simplifying this gives \(25x^2 + 10x\). Now, the equation is \(25x^2 + 10x = 8\).
02

Rearrange the Equation

To solve the quadratic equation, move all terms to one side, setting the equation to zero. Subtract 8 from both sides: \(25x^2 + 10x - 8 = 0\).
03

Use the Quadratic Formula

The quadratic equation is now in the form \(ax^2 + bx + c = 0\), with \(a = 25\), \(b = 10\), and \(c = -8\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions for \(x\).
04

Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\) to determine the solution's nature. Compute \(10^2 - 4 \times 25 \times (-8) = 100 + 800 = 900\). A positive discriminant means two real and distinct solutions.
05

Solve for x

Substitute all known values into the quadratic formula: \(x = \frac{-10 \pm \sqrt{900}}{50}\). Compute the square root, which gives \(x = \frac{-10 \pm 30}{50}\). This results in two possible solutions: \(x = \frac{20}{50} = \frac{2}{5}\) and \(x = \frac{-40}{50} = -\frac{4}{5}\).

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
The quadratic formula is a powerful tool used to solve quadratic equations, which are equations in the form \( ax^2 + bx + c = 0 \). The quadratic formula itself is given by:
  • \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
It allows us to find the values of \( x \) that make the quadratic equation true. Here, \( a \), \( b \), and \( c \) are coefficients from the equation, and \( \pm \) indicates that there might be two possible values for \( x \). The formula derives from completing the square method and can solve any quadratic equation as long as the square root and division calculations are possible.
Discriminant
The discriminant is an essential part of the quadratic formula, located under the square root: \( b^2 - 4ac \). It reveals critical information about the nature of the roots of the quadratic equation. By evaluating the discriminant, you can quickly understand the following:
  • If the discriminant is positive, like in our example where it equaled 900, the equation has two distinct real solutions.
  • If the discriminant is zero, there is exactly one real solution, known as a repeated or double root.
  • If the discriminant is negative, the equation has no real solutions but rather two complex solutions.
This makes the discriminant a valuable first step when analyzing the behavior of quadratic equations.
Real Solutions
Real solutions are the values of \( x \) that solve the quadratic equation, provided they are real numbers. In the process we followed, the discriminant was positive, indicating two real solutions:
  • \( x = \frac{2}{5} \)
  • \( x = -\frac{4}{5} \)
Real solutions exist when the discriminant is zero or positive. These solutions can be easily found by substituting back into the original equation to verify their correctness. Real solutions are commonly plotted on a graph where they appear as the points where the quadratic function intersects the x-axis.
Polynomial Expansion
Polynomial expansion involves rewriting or simplifying expressions, particularly ensuring terms are simplified into a more manageable form. In our exercise, the expression \( 5x(5x + 2) \) was expanded to become \( 25x^2 + 10x \) by using the distributive property.
  • \( 5x \cdot 5x = 25x^2 \)
  • \( 5x \cdot 2 = 10x \)
This procedure transforms a product into a sum, making it easier to rearrange and solve equations. Polynomial expansion is a crucial foundational skill in algebra that facilitates the solving of more complex algebraic equations, such as our quadratic equation solving task.

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 Problems \(71-88\), set up an equation and solve each problem. (Objective 4) Explain how you would solve \((x+2)(x+3)=(x+2)\) \((3 x-1)\). Do you see more than one approach to this problem?

\(x^{2}+8 x=20\)

Suppose that one leg of a right triangle is 7 meters longer than the other leg. The hypotenuse is 1 meter longer than the longer leg. Find the lengths of all three sides of the right triangle.

For Problems \(13-40\), factor each polynomial completely. Indicate any that are not factorable using integers. (Objective 2) $$ 4 x^{2}-28 x y+49 y^{2} $$

For Problems \(71-88\), set up an equation and solve each problem. (Objective 4) In an office building, a room contains 54 chairs. The number of chairs per row is three less than twice the number of rows. Find the number of rows and the number of chairs per row.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

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.