How to Solve These Quadratic Equations?

Summary

In this educational video, the instructor demonstrates how to solve quadratic equations using the method of extracting square roots. They provide a step-by-step approach to simplifying equations and isolating the variable, offering two solutions for each example. The video is aimed at grade 10 students, with the instructor, Ram Islam, engaging viewers with clear explanations and practical examples. New subscribers are encouraged to like and subscribe for more educational content.

Takeaways

• 📚 The video is an educational tutorial focused on solving quadratic equations.
• 🔍 The first example involves simplifying the equation by eliminating constants and coefficients.
• 📉 The method of transposing terms to one side of the equation is demonstrated to isolate the variable.
• 🔢 The process of dividing both sides by a coefficient to simplify the equation is explained.
• 🆚 The use of square roots to solve for the variable is introduced, including both positive and negative roots.
• 📝 The importance of considering both the positive and negative square roots when solving is emphasized.
• ✂️ The concept of canceling out terms in the equation to simplify it further is shown.
• 📉 The second example involves dealing with fractional forms and demonstrates how to eliminate the fraction.
• 🔄 The tutorial covers multiplying both sides of the equation by the denominator to clear the fraction.
• 📌 The final step in both examples is to isolate the variable and solve for its possible values.
• 👨‍🏫 The presenter, MIDI turgon Ram Islam, encourages viewers to like and subscribe for more educational content.

Q & A

What is the main topic of the video?

The main topic of the video is how to solve quadratic equations.

What method is suggested for solving the first quadratic equation in the video?

The method suggested for solving the first quadratic equation is by extracting square roots.

How does the video suggest to handle the negative 42 in the first equation?

The video suggests transposing the negative 42 to the other side of the equation to make it positive 42.

What is the next step after transposing the negative 42 in the first equation?

The next step is to eliminate the positive six by dividing both sides of the equation by six.

What does the video suggest to do after simplifying the equation to 'five plus X squared equals seven'?

The video suggests multiplying both sides of the equation by the conjugate to extract the square roots.

How does the video handle the extraction of square roots when 7 is not a perfect square?

The video suggests taking the positive and negative square root of 7 and then isolating X by transposing 5 to the other side of the equation.

What are the two possible values of x for the first equation according to the video?

The two possible values of x are (positive square root of 7) - 5 and (negative square root of 7) - 5.

What is the approach for solving the second equation in the video?

The approach for the second equation is to first transpose negative 13 to the other side, then multiply both sides by the denominator to eliminate it, and finally extract the square roots.

How does the video suggest to simplify the equation after multiplying by the denominator in the second problem?

The video suggests simplifying to '2x plus 3 squared equals 39' and then extracting the square roots to solve for x.

What are the two possible values of x for the second equation according to the video?

The two possible values of x are (positive square root of 39 - 3) / 2 and (negative square root of 39 - 3) / 2.

Who is the presenter of the video and what is their closing remark?

The presenter of the video is MIDI turgon, Ram Islam, and their closing remark is a reminder to like and subscribe for updates.

Outlines

00:00

📚 Solving Quadratic Equations by Square Roots

This paragraph introduces the process of solving quadratic equations using the square root method. The teacher begins by presenting the first problem, which involves an equation with a term 'six times the quantity of 5 plus X squared minus 42'. The initial step is to eliminate the negative term by transposing it to the other side of the equation, resulting in 'six times the quantity of five plus X squared equals 42'. The next step is to eliminate the coefficient of the quadratic term by dividing both sides by six, simplifying the equation to 'five plus X squared equals seven'. The teacher then demonstrates how to apply the square root method to solve for X, resulting in two potential solutions: 'X equals plus or minus the square root of seven minus five'. The paragraph concludes with a note on how to present the solutions in different forms if required by the teacher.

05:01

🔍 Step-by-Step Solution for Quadratic Equations

The second paragraph continues the theme of solving quadratic equations, focusing on a step-by-step approach for a specific problem. The equation '2X plus 3 squared equals negative 13' is presented, and the teacher explains how to transpose the negative term to the other side, resulting in '2X plus 3 squared equals 13'. The teacher then demonstrates the process of eliminating the denominator by multiplying both sides by three, leading to '2X plus 3 squared equals 39'. The square root method is applied next, yielding two potential solutions: 'X equals plus or minus the square root of 39 minus 3 divided by 2'. The teacher emphasizes the importance of considering these as the final answers and provides guidance on how to present alternative solutions if necessary. The paragraph ends with a reminder for viewers to subscribe to the channel for updates.

Keywords

💡Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically written in the form ax^2 + bx + c = 0. In the video, the teacher introduces the topic of solving quadratic equations, which is the main theme. The script presents two examples of such equations, demonstrating the process of solving them.

💡Extracting Square Roots

Extracting square roots is a method used to solve quadratic equations when they are in a form that allows for direct application of the square root to both sides of the equation. In the script, this method is used to simplify the equation by taking the square root of both sides, which is a key step in solving the first example.

💡Transposing

Transposing in the context of algebra involves moving terms from one side of an equation to the other to isolate the variable. The script mentions transposing negative 42 to the other side of the equation to simplify it, which is an essential step in the process of solving quadratic equations.

💡Dividing by Coefficient

Dividing by the coefficient is a technique used to simplify the equation by making the coefficient of the variable term equal to one. In the video script, the teacher divides both sides by six to eliminate the coefficient of the term involving 'X squared', making the equation easier to solve.

💡Perfect Square

A perfect square is a number that can be expressed as the square of an integer. In the script, the term '7' is mentioned as not being a perfect square, which means its square root cannot be simplified to an integer, and therefore the square root of seven must be used in the solution.

💡Quadratic Terms

Quadratic terms refer to the terms in a quadratic equation that involve the variable raised to the second power. In the script, the teacher talks about eliminating other terms to leave only the quadratic term on one side of the equation, which is a standard approach in solving quadratic equations.

💡Radicand

The radicand is the number or expression under a radical sign, in this case, the square root. In the script, the teacher refers to the radicand when explaining the process of extracting square roots from the equation, which is crucial for finding the solutions.

💡Factoring

Factoring is the process of breaking down a polynomial into a product of its factors. Although not explicitly mentioned in the script, the concept is implied when the teacher discusses the possibility of factoring the negative sign in the second example, which is a method to find the solutions to quadratic equations.

💡Fractional Forms

Fractional forms refer to the representation of numbers as fractions. In the second example of the script, the teacher mentions dealing with fractional forms, which is part of the process of solving the quadratic equation involving a fraction.

💡Solving for X

Solving for 'X' means finding the values of the variable that satisfy the equation. The script provides a step-by-step guide on how to find the values of 'X' for two different examples of quadratic equations, which is the ultimate goal of the video.

💡Properties of Equality

The properties of equality state that you can perform the same operation on both sides of an equation without changing the equality. In the script, the teacher uses these properties to add, subtract, and divide terms to manipulate the equation and solve for 'X'.

Highlights

Introduction to the topic of solving quadratic equations.

Explanation of different methods to solve quadratic equations.

Presentation of the first problem involving the equation 6(5 + x)^2 - 42.

Strategy to eliminate negative 42 by transposing it to the other side of the equation.

Simplification of the equation to resemble the form x^2 = k.

Division of both sides by 6 to isolate the quadratic term.

Use of the square root extraction method to solve the simplified equation.

Handling of the imperfect square by taking the positive and negative square roots.

Transposition of the constant term to isolate the variable.

Final simplification to find the two possible values of x.

Introduction of the second problem involving fractional forms.

Step-by-step guide on solving the second problem with fractional coefficients.

Multiplication of both sides by the denominator to eliminate the fraction.

Application of the square root extraction method to the new equation.

Isolation of the variable by transposing terms and dividing by the coefficient.

Presentation of the two possible solutions for x in the second problem.

Emphasis on the properties of equality in solving quadratic equations.

Conclusion summarizing the methods for solving quadratic equations in grade 10.

Encouragement for new subscribers to like and subscribe for updates.

Transcripts

hi guys it's me teacher gone in today's

video we will talk about how to solve

quadratic equations

so melaton different versions of this

kind of topic pero now

when it comes to solving quadratic

equations

so without further Ado let's do this

topic

so what we have here is the first

problem

we are given the equation

six times the quantity of 5 plus X

raised to the second power

minus 42 and to give you an idea

what we're going to use here is solving

quadratic equations by extracting the

square roots

and to solve this problem first we will

eliminate negative 42

guys you can add 42 both sides of

decoration but to make it easier for us

it will just transpose negative 42 to

the other side of the equation

that would make it positive for two so

what will happen our new equation is

six

times the quantity of five

plus X

squared

is equal to from negative it will become

just 42. then after that

guys is to follow this pattern we need

to make it

looks like a x squared

is equal to K that but

quadratic terms left side and then

constant on the other side so what's

next here is that we need to eliminate

positive six

so to eliminate that is

divide both sides

by six in this case you can cancel out

six

we can cancel out six

what we mean here is five

plus X

raised to the second power

is equal to 42 divided by 6 is

equal to

seven now

in your case maybe you will you would

think that we need to expand this one we

will try to

multiply 5 plus x times five plus X

perilous of using extracting the square

roots

connecting square roots now both sides

of the equation

credits and this one don't forget the

positive and negative and by the way

comment section

your radical and the Power of Two

invisible index not to

nah you may property if the index

and exponent of the radicand are equal

we can simply cancel it out

so what we have here is five

plus X

is equal to since 7 is not a perfect

square and well anti-factors

the seven

it only mean as positive negative

square root of seven

what's next is

five

by transposing it to the other side of

the equation so what we have now is

simply copier X

is equal to positive negative

squared off

seven then it will become minus 5. hang

on

um you can consider this one as an

answer key

this is simplified by you negative seven

so minus five okay acceptable in the end

but if your teacher or Professor asks

you for little different solutions you

can separate the two solutions

your first solution use the positive 7

square root of seven

that is square root of seven and then

copy

negative five so this is the first value

of x

now for the second solution which is the

x sub 2

we are done with the positive negative

you have negative square root of

seven

minus five foreign

if your teacher wants to factor the

negative sign

again negative times square root of

seven

plus five this one is possible so these

are the possible values of x

so Mejo

um

the properties of equality now let's

continue with item number two

in number two fractional forms are

better don't worry about it

I will teach you step by step on how to

solve this one

same thing that we learn to go in okay

x squared

is equal to K

right now

so left side

negative 13.

is equivalent to x squared okay

we will transpose negative 13 to the

other side of the equation

this will become

2X

plus 3

raised to the second power

over three transpose negative 13 to the

other side

it will become

positive 13. and here

in denominator three so what we need to

do is to eliminate it by

multiplying both sides of the equation

by this denominator

it will be canceled out is 2X

plus 3

raised to the second power and on the

other side 13 times 3 is

30

9.

so what's next

extract the square roots

x squared 39 is X is K

get a square root of this one and then

positive negative

cancel cancel

you have 2X

plus 3

is equal to

positive negative square root of 39.

okay

when is transpose this to the other side

of the equation

to become

2x is equal to positive negative square

root of

39

minus three

and lastly

and as a sold for X Style

divide both sides by two

cancel cancel

your

cash up your X move that internal

contact

your X

is equal to

positive negative square root of

39

minus 3 over

2. actually guys

you should consider this one as final

answer because a

uh factors

Solutions you can provide one

your x sub 1 here

square root of 39

minus 3 over

as the first value of x

for the second value of x

negative domain

negative square root of

39

minus 3 over 2 and these are the

possible values of x

so yo guys

on how to solve quadratic equations

on how to solve quadratic equations

for grade 10. so guys if you're new to

my channel

don't forget to like And subscribe

button

for you to be updated certain latest

uploads again it's MIDI turgon

Ram Islam

bye