/*! 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 30 Solve. $$ 5 x(x-1)-7=4 x(x-2)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 30

Solve. $$ 5 x(x-1)-7=4 x(x-2) $$

Short Answer

Expert verified
The solutions are \( x = \frac{-3 + \sqrt{37}}{2} \) and \( x = \frac{-3 - \sqrt{37}}{2} \).

Step by step solution

01

- Expand both sides of the equation

First, expand the left and right sides of the equation.For the left side: \[ 5x(x-1) - 7 \rightarrow 5x^2 - 5x - 7 \]For the right side: \[ 4x(x-2) \rightarrow 4x^2 - 8x \]Now the equation looks like: \[ 5x^2 - 5x - 7 = 4x^2 - 8x \]
02

- Move all terms to one side of the equation

To simplify the equation, subtract all terms on the right side from both sides of the equation: \[ 5x^2 - 5x - 7 - 4x^2 + 8x = 0 \]Combine like terms: \[ 5x^2 - 4x^2 + 8x - 5x - 7 = 0 \]So, the equation becomes: \[ x^2 + 3x - 7 = 0 \]
03

- Solve the quadratic equation

Use the quadratic formula to solve for x. The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In this equation, a = 1, b = 3, and c = -7. Plug these values into the formula: \[ x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-7)}}{2 \cdot 1} \]Simplify under the square root: \[ x = \frac{-3 \pm \sqrt{9 + 28}}{2} \]Further simplify: \[ x = \frac{-3 \pm \sqrt{37}}{2} \]So, the solutions are: \[ x = \frac{-3 + \sqrt{37}}{2} \] and \[ x = \frac{-3 - \sqrt{37}}{2} \]

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 for solving quadratic equations, which are equations in the form of \[ ax^2 + bx + c = 0 \]. The formula is written as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. To use the formula, first identify the coefficients \( a \), \( b \), and \( c \) from the equation. Next, substitute these values into the formula. Make sure to be careful with the order of operations, especially under the square root. The quadratic formula provides two solutions: one with the plus sign and one with the minus sign.
expanding equations
Expanding equations is the process of multiplying out expressions to remove parentheses. For example, if you have the term \( 5x(x-1) \), you distribute \( 5x \) to both \( x \) and \( -1 \). This results in \( 5x^2 - 5x \). Similarly, for \( 4x(x-2) \), you distribute \( 4x \) to both \( x \) and \( -2 \), resulting in \( 4x^2 - 8x \). This step helps in simplifying equations, preparing them for further operations such as combining like terms.
combining like terms
Combining like terms involves adding or subtracting terms that have the same variables raised to the same powers. For instance, in the expression \( 5x^2 - 5x - 7 - 4x^2 + 8x \), you can combine \( 5x^2 \) and \( -4x^2 \) to get \( x^2 \). Likewise, combine \( -5x \) and \( 8x \) to get \( 3x \). The constant term \( -7 \) does not change. Combining like terms simplifies the equation, making it easier to solve.
simplifying expressions
Simplifying expressions involves reducing them to a form that is easier to work with. This may include expanding equations, combining like terms, and moving all terms to one side of the equation. For instance, after expanding and combining like terms, the equation \( 5x(x-1) - 7 = 4x(x-2) \) simplifies to \( x^2 + 3x - 7 = 0 \). Simplification makes it easier to apply further solving techniques like 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

For each equation under the given condition, (a) find \(k\) and (b) find the other solution. $$ k x^{2}-2 x+k=0 ; \text { one solution is }-3 $$

Write an equation for a function having a graph with the same shape as the graph of \(f(x)=\frac{3}{5} x^{2},\) but with the given point as the vertex. $$ (2,6) $$

Let \(f(x)=x^{2} .\) Find \(x\) such that \(f(x)=11\).

Let \(F(t)=(t+4)^{2} .\) Find \(t\) such that \(F(t)=13\)

Find (a) the maximum or minimum value and (b) the \(x\) - and \(y\) -intercepts. Round to the nearest hundredth. $$ g(x)=0.45 x^{2}-1.72 x+12.92 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision 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.