Solving a rational Equation

Summary

This video covers the process of solving a rational equation by eliminating fractions through multiplying by the least common denominator (LCD). The instructor explains how to identify the LCD and eliminate fractions, leading to a simplified quadratic equation. The equation is then factored and solved using the zero product property, resulting in two possible solutions. The walkthrough offers clear steps and guidance for solving rational equations, particularly for equations that involve quadratics.

Takeaways

• 🔢 Identifying the LCD (Least Common Denominator) is the first step to simplify the rational equation.
• ✖️ The LCD in this case is determined to be 2x, which divides into each of the denominators.
• 🧮 Multiply every term in the equation by the LCD to eliminate fractions.
• ➗ The denominators divide out after multiplying by the LCD, simplifying the equation to 2 + x^2 = x + 4.
• 📐 The equation becomes quadratic after simplifying, requiring factoring techniques.
• 📉 To factor, get all terms to one side and set the equation equal to zero.
• ✏️ Rearranging gives the equation x^2 - x - 2 = 0, which needs to be factored.
• 🔗 The equation factors into (x - 2)(x + 1) = 0.
• 🟰 Using the zero product property, solve the two linear equations: x - 2 = 0 and x + 1 = 0.
• ✅ The final solutions to the rational equation are x = 2 and x = -1.

Q & A

What is the first step in solving the rational equation in the script?

The first step is to determine the Least Common Denominator (LCD) of all the fractions and multiply the entire equation by that LCD to eliminate the fractions.

How is the LCD determined in this specific equation?

The LCD is determined by looking at the denominators: x, 2, and 2x. The smallest term that all of them divide into is 2x.

What happens when the equation is multiplied by the LCD?

Multiplying by the LCD eliminates the denominators, leaving an equation without fractions.

What does the equation become after multiplying by the LCD?

After multiplying by the LCD (2x), the equation becomes 2 + x² = x + 4.

What type of equation does the script mention it becomes after eliminating the fractions?

The equation becomes a quadratic equation after eliminating the fractions.

What is the next step after identifying the equation as a quadratic?

The next step is to move all the terms to one side of the equation, setting it equal to zero, to prepare for factoring.

How is the quadratic equation factored in this case?

The quadratic equation x² - x - 2 = 0 is factored into (x - 2)(x + 1) = 0.

What is the zero product property, and how is it applied here?

The zero product property states that if a product of two factors equals zero, then at least one of the factors must be zero. It's applied by setting each factor (x - 2) and (x + 1) equal to zero.

What are the solutions to the quadratic equation after applying the zero product property?

The solutions are x = 2 and x = -1.

Why does the speaker subtract x and 4 during the process?

The speaker subtracts x and 4 to move all terms to one side of the equation, making it easier to rewrite the quadratic equation in standard form for factoring.

Outlines

00:00

📊 Solving Rational Equations with LCD

The speaker begins by addressing a rational equation and reassures that eliminating fractions can be simple. The first step is identifying the least common denominator (LCD) for all the fractions involved, which in this case is 2x. The speaker explains how each fraction divides into the LCD, allowing the elimination of fractions by multiplying every term by the LCD. This leads to a simplified equation without fractions.

🔢 Simplifying to a Quadratic Equation

Once the fractions are eliminated, the speaker points out that the new equation is quadratic. They emphasize the importance of recognizing this and shifting from isolating variables to factoring. The goal is to rearrange the equation so that all terms are on one side, setting the equation equal to zero. This step is crucial for factoring and applying the zero product property later.

➗ Factoring the Quadratic Expression

The quadratic equation is rewritten in descending order: x² - x - 2 = 0. The speaker explains how to factor the equation by identifying two numbers that multiply to give -2 and add to give -1. The factors are (x - 2)(x + 1), setting up the application of the zero product property, which is essential for solving the equation.

✅ Solving for the Variable

With the factored equation, the speaker uses the zero product property, setting each factor equal to zero. This results in two simple linear equations: x - 2 = 0 and x + 1 = 0. Solving these yields the two solutions for x: x = 2 and x = -1. The speaker concludes by noting that this rational equation has two solutions.

Keywords

💡Rational Equation

A rational equation is an equation that involves at least one rational expression, where the numerator and/or denominator contain variables. In the video, the speaker discusses solving a rational equation by eliminating the fractions involved through a technique known as multiplying by the least common denominator (LCD).

💡LCD (Least Common Denominator)

The least common denominator is the smallest number or expression that each of the denominators in a set of fractions can divide into without leaving a remainder. In the video, the speaker identifies the LCD as 2x, allowing the elimination of fractions by multiplying each term by this value, simplifying the equation.

💡Denominator

The denominator is the bottom part of a fraction that shows how many equal parts the whole is divided into. The video discusses eliminating the denominators in the rational equation by multiplying through by the least common denominator (LCD), effectively simplifying the equation.

💡Multiply Every Term

Multiplying every term by the least common denominator is a strategy used to eliminate fractions from an equation. The speaker highlights this method, showing how it simplifies each term by canceling out the denominators, making the equation easier to solve.

💡Quadratic Equation

A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where the highest power of the variable is squared. In the video, after eliminating the fractions, the speaker notes that the resulting equation becomes quadratic, prompting the need for factoring to find the solution.

💡Zero Product Property

The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. This property is used in the video after factoring the quadratic equation, allowing the speaker to set each factor equal to zero and solve for the variable.

💡Factoring

Factoring is the process of breaking down a polynomial into simpler terms, or factors, that when multiplied together give the original polynomial. The speaker uses factoring to solve the quadratic equation x² - x - 2 = 0, breaking it down into (x - 2)(x + 1) = 0.

💡Inverse Operations

Inverse operations are operations that undo each other, such as addition and subtraction or multiplication and division. In the video, after applying the zero product property, the speaker uses inverse operations to solve for x, isolating the variable to find the solutions.

💡Linear Equation

A linear equation is an equation where the variable is raised to the first power, resulting in a straight-line graph when plotted. The speaker simplifies the quadratic equation into two linear equations, x - 2 = 0 and x + 1 = 0, and solves for x using basic algebra.

💡Solutions

Solutions are the values of the variable that satisfy an equation. In the video, after factoring the quadratic equation and applying the zero product property, the speaker finds two solutions, x = 2 and x = -1, which satisfy the original rational equation.

Highlights

Introduction to solving rational equations by determining the LCD.

Emphasis on eliminating fractions by multiplying by the LCD.

Explanation of determining the smallest term that all denominators divide into.

The LCD is identified as 2x because it divides into each denominator.

Multiplying all terms by the LCD to eliminate fractions.

Step-by-step breakdown of how each denominator divides out, simplifying the equation.

Resulting equation after eliminating fractions: 2 + x² = x + 4.

Recognizing the equation as a quadratic equation.

Subtraction of x from both sides to begin isolating terms.

Rewriting the equation in descending order to prepare for factoring.

Factoring the quadratic equation: (x - 2)(x + 1) = 0.

Application of the zero-product property to solve the factored equation.

Solving the two linear equations: x - 2 = 0 and x + 1 = 0.

The two solutions to the rational equation are x = 2 and x = -1.

Conclusion of solving the rational equation with two solutions and review of the process.

Transcripts

okay

so here we go again we have a nice

lovely rational equation and again to

kind of soul

quiet down your fears we can eliminate

all of these fractions by doing one

simple thing that is first of all

determining what the lcd is each one of

our fractions and then

multiplying by that value so i look up

here i have an x a 2 and a 2x

so what i need to do is determine what

is the smallest term that each and every

one of those values divide into well

if i say my lcd is 2x we know that

obviously 2x divides into 2x 2 divides

into 2x x times and x divides in the 2x

two times so therefore each one of my

denominators divide into my lcd so now

to eliminate my l my denominators i'm

going to multiply every single term

by

my lcd

and the way that this eliminates my

fractions now is because now i have a

part in each denominator that i know

divides into lcd here my x divide out to

1.

here my twos divide out to 1. and here

my 2x divides into the 2x so it divides

out therefore leave me with an equation

of 2

plus

x squared

equals x plus 4.

now i simply need to get now i notice

that this is a quadratic now right

so

rather than just saying oh i just need

to isolate the variable we need to look

into some factoring techniques so when

we have a quadratic we want to get all

of our variables to the same side and

set it equal to zero the reason why we

want to set it equal to zero because we

want to factor it and try to apply the

zero product property so what i'll do

now is i'll subtract an x

and so therefore i get two

plus

x squared minus x

equals 4. then i'll subtract 4 here so

now i'm just going to rewrite this in

descending order so i'll have x squared

minus x

minus two equals zero

now i need to determine can i factor

this any four what two numbers

multiplied giving me negative two but

then i had to give me a negative one

well we could say that's going to be x

minus two

times x plus one equals zero and now

that i have a product equal to zero i

can apply the zero product property

which says x minus two equal to zero or

x plus one equals zero and now what i've

done is i've created two linear

equations that equal zero so now i can

simply use my inverse operations to

solve so x equals 2

and x equals negative 1. so therefore

this rational expression a rational

equation has two solutions thanks

you