Solving a rational Equation
Summary
TLDRThis 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
📊 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.
Mindmap
Keywords
💡Rational Equation
💡LCD (Least Common Denominator)
💡Denominator
💡Multiply Every Term
💡Quadratic Equation
💡Zero Product Property
💡Factoring
💡Inverse Operations
💡Linear Equation
💡Solutions
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
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