Solving Quadratic Equations by Extracting the Square Roots by @MathTeacherGon

Summary

In this educational video, the host, Teacher Gone, explains how to solve quadratic equations by extracting square roots, following up on a previous lesson about factoring. The video demonstrates solving equations like 4x^2 - 9 = 0, first by factoring and then using the square root method. The host guides viewers through the process of transposing constants, isolating x^2, and taking square roots, including handling perfect squares and non-perfect squares. The video also covers solving equations with positive and negative results, leading to real and imaginary solutions. The host promises a part two for more examples and encourages viewers to subscribe for updates.

Takeaways

• 📚 The video focuses on solving quadratic equations by extracting square roots, following a discussion on factoring in a previous video.
• 🔢 The standard form for solving quadratic equations is ax^2 + bx + c = 0, and the script starts with an example where a = 4 and c = -9.
• ➗ The process involves factoring the equation first, if possible, and then extracting square roots to find the solutions.
• 🔄 When using the square root method, the equation is rearranged to the form ax^2 = c, and then the square root of both sides is taken.
• 📉 For equations like 4x^2 - 9 = 0, the constant term is moved to the other side, and the square root of each side is extracted to find x.
• 🔠 The solutions to the example 4x^2 - 9 = 0 are x = 3/2 and x = -3/2, which are derived both by factoring and by extracting square roots.
• 🔢 The video demonstrates how to handle perfect squares under the square root, such as in x^2 - 100 = 0, leading to solutions x = 10 and x = -10.
• 📝 The script also covers non-perfect squares, showing how to rationalize the denominator and simplify the solutions, as seen in 9x^2 = 8.
• 🌐 The video includes an example with an irrational number under the square root, resulting in solutions involving √2 over √9, or simplified as 2√2/3 and -2√2/3.
• 🚫 The method is applied to equations with positive and negative values under the square root, including an example where the square root of a negative number introduces imaginary units i.

Q & A

What is the main topic of the video?

The main topic of the video is solving quadratic equations by extracting square roots.

What is the first method discussed for solving quadratic equations in the video?

The first method discussed for solving quadratic equations is factoring.

What is the standard form of a quadratic equation mentioned in the video?

The standard form of a quadratic equation mentioned in the video is ax^2 + bx + c = 0.

How is the equation 4x^2 - 9 solved using factoring in the video?

The equation 4x^2 - 9 is solved by recognizing it as a difference of squares, factoring it into (2x + 3)(2x - 3), and then setting each factor equal to zero to find the solutions x = -3/2 and x = 3/2.

What is the pattern for extracting square roots in solving quadratic equations?

The pattern for extracting square roots in solving quadratic equations is ax^2 = c, where you transpose the constant term to the other side and then take the square root of both sides.

How is the equation x^2 - 100 = 0 solved using the square root method in the video?

The equation x^2 - 100 = 0 is solved by transposing -100 to the other side to get x^2 = 100, then taking the square root of both sides to find the solutions x = ±10.

What is the solution to the equation 9x^2 = 8 using the square root method as shown in the video?

The solution to the equation 9x^2 = 8 is found by dividing both sides by 9 to get x^2 = 8/9, then taking the square root to find x = ±(2/3)√2.

How is the equation 2x^2 = 3 solved using the square root method in the video?

The equation 2x^2 = 3 is solved by dividing both sides by 2 to get x^2 = 3/2, then taking the square root to find x = ±√(3/2).

What happens when the constant term in a quadratic equation is negative when using the square root method?

When the constant term in a quadratic equation is negative, the square root of a negative number results in an imaginary number, represented by 'i' in the solutions.

What are the solutions to the equation x^2 + 25 = 0 using the square root method as explained in the video?

The solutions to the equation x^2 + 25 = 0 are x = ±5i, where 'i' represents the imaginary unit.

What is the advice given to viewers at the end of the video regarding the channel?

The advice given to viewers at the end of the video is to like, subscribe, and hit the Bell button to stay updated with the latest uploads.

Outlines

00:00

📘 Introduction to Solving Quadratic Equations

In this segment, the instructor introduces the topic of solving quadratic equations by extracting square roots. The video builds upon previous lessons on factoring and introduces a new method for solving such equations. The instructor reminds viewers of the standard form of a quadratic equation, ax^2 + bx + c = 0, and uses the example of 4x^2 - 9 = 0 to demonstrate solving by factoring. The equation is factored into (2x + 3)(2x - 3) = 0, leading to the solutions x = -3/2 and x = 3/2. The segment transitions into explaining how to use the square root extraction method for equations of the form ax^2 = c, where b = 0.

05:02

🔍 Detailed Explanation of Square Root Extraction Method

This paragraph delves into the square root extraction method for solving quadratic equations. The instructor uses the equation 4x^2 - 9 = 0 to illustrate the process. The constant term is moved to the other side, resulting in 4x^2 = 9. The square root of both sides is taken, yielding 2x = ±3. After simplifying, the solutions x = ±3/2 are obtained, which match the solutions from factoring. The instructor then moves on to additional examples, demonstrating the method with equations like x^2 - 100 = 0, leading to solutions x = ±10, and 9x^2 = 8, resulting in x = ±2√2/3. The process is explained step by step, with emphasis on the importance of considering both the positive and negative square roots.

10:03

📚 Concluding Quadratic Equation Solutions and Future Content

In the final paragraph, the instructor concludes the discussion on solving quadratic equations using the square root extraction method. An example with an imaginary number is presented: x^2 + 25 = 0, which leads to the solution x = ±5i, where i is the imaginary unit. The instructor emphasizes that this method will be accepted by teachers and assures viewers that they have learned valuable skills. A promise is made to create a part two of the video to further discuss solving quadratic equations by extracting square roots. The instructor invites new viewers to like, subscribe, and enable notifications for the latest uploads, and signs off with a farewell.

Keywords

💡Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically in the form ax² + bx + c = 0. In the video, the teacher focuses on solving quadratic equations by factoring and extracting square roots, emphasizing different methods to find the solutions.

💡Factoring

Factoring is a method used to solve quadratic equations by expressing the quadratic as a product of its factors. In the video, the teacher demonstrates factoring using the example 4x² - 9 = 0, showing how to rewrite it as (2x + 3)(2x - 3) = 0, which then leads to finding the roots.

💡Extracting Square Roots

Extracting square roots is another method to solve quadratic equations where one isolates the x² term and takes the square root of both sides. The teacher explains this approach after solving the equation by factoring, using examples like 4x² = 9 and x² - 100 = 0.

💡Difference of Two Squares

The difference of two squares is a specific form of factoring where a quadratic takes the form a² - b². The teacher uses this concept to solve 4x² - 9 = 0, breaking it down as (2x + 3)(2x - 3) = 0.

💡Standard Form

Standard form refers to the structure of a quadratic equation, ax² + bx + c = 0. The video reminds viewers that quadratic equations should be in standard form before solving, as shown in examples like 4x² - 9 = 0.

💡Positive and Negative Roots

Quadratic equations often have two solutions: one positive and one negative. This concept is emphasized throughout the video, such as when solving x² = 100, leading to solutions of x = ±10.

💡Transpose

Transposing refers to moving a term from one side of the equation to the other, changing its sign in the process. The teacher repeatedly uses this technique, such as when shifting constants like -9 to the other side of the equation to simplify it.

💡Imaginary Numbers

Imaginary numbers arise when dealing with the square root of a negative number. The teacher introduces this concept when solving x² = -25, explaining how the square root of -25 results in an imaginary number, represented as 5i.

💡Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. In the video, the teacher explains how to extract square roots from both sides of an equation, such as finding the square root of 9 to get 3, and highlights the importance of considering both positive and negative roots.

💡Perfect Squares

A perfect square is a number that has an integer as its square root. The teacher demonstrates this using examples like 100 and 9, where the square roots are whole numbers (10 and 3, respectively), simplifying the process of solving the quadratic equations.

Highlights

Introduction to solving quadratic equations by extracting square roots.

Review of solving quadratic equations by factoring.

Explanation of converting equations to standard form ax^2 + bx + c = 0.

Factoring the equation 4x^2 - 9 = 0 as (2x + 3)(2x - 3) = 0.

Solving the factored equation to find x = ±3/2.

Method for extracting square roots from quadratic equations.

Pattern for equations of the form ax^2 = c where b = 0.

Transposing constants to one side to isolate x^2.

Extracting square roots to solve 4x^2 = 9 resulting in x = ±3/2.

Solving x^2 - 100 = 0 to find x = ±10.

Solving 9x^2 = 8 using square roots and division.

Result for 9x^2 = 8 is x = ±(2√2)/3.

Solving 2x^2 = 3 by dividing and extracting square roots.

Final answer for 2x^2 = 3 is x = ±√(3/2).

Solving x^2 + 25 = 0 using imaginary numbers.

Result for x^2 + 25 = 0 is x = ±5i.

Encouragement to like, subscribe, and hit the Bell button for updates.

Transcripts

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

video we will talk about solving

quadratic equations by extracting the

square roots from our previous video we

talked about how to solve quadratic

Creations by factoring and also the

introduction about quadratic equations

without further ado

let's do this topic

so before we start diving into the

exactly the square roots let me remind

you first how to use

solving quadratic equations by factoring

and this equation is that we have 4x

squared minus 9. we will try to solve it

first by factoring and then later on we

will use the extracting discrete root

now remember

that in Factory you need to

convert the equation in standard form

ax squared

plus BX

plus c

is equal to zero and this equation is

already in this form we have 4 x squared

minus 9 is equal to zero Now by

factoring this one is under difference

of two squares meaning we can Factor

this out as

2x plus 3

times 2x minus 3.

okay the square root of 4x squared is 2X

square root of 9 is 3 and after

factoring it out

we need to equate each factor by zero so

we have two X plus 3

is equal to zero and the other is 2X

minus 3 is equal to zero transpose this

to the other side

it will become 2X

is equal to negative three

then divide voltage by 2

and as you can see cancel cancel

your X is equal to negative three over

two now what about the other

equation

transpose this equation it will become

2X

is equal to

positive 3

divide voltage by two

cancel cancel your X is equal to three

over two to sum it up

the solution of this equation are

positive and negative three over two now

how are we going to use

the extracting the square roots

in this kind of equation in extracting

the square root we need to follow this

pattern

we have

a x squared

is equal to

C we're in here B is equal to it's not

equal to

zero or equal to zero

now for this type of equation as you can

see we have here four

x squared minus 9 is equal to zero we

need to follow this pattern ax squared

is equal to zero what will happen here

is we need to transpose 9 to the other

side

because this constant is on the other

side

transpose that into it will become four

x squared

is equal to from negative 9 it will

become

positive 9. so after that

again we're starting to use the

extracting the square root

after following this pattern we can use

or extract the square roots how get the

square root of this

this will become

2X

and this one

square root of this

again positive and negative so it'll

become

positive negative square root of nine is

three but we're not yet done because

so what will happen

divide both sides by 2 divide by 2

cancel cancel

your X or the solutions are

positive negative

3 over 2. and as you can see

we have the same answer from example

number one using Factory

now let's move on to other examples

where in we will

use solving equations by extracting the

square roots

so we still have

different examples here

so what will happen is that we will

start with number one

we have x squared minus 100 is equal to

zero

follow this pattern

ax squared

is equal to C meaning this negative 100

must be transposed

to the other side it will become

x squared

is equal to

100 and after that

get a square root

get a squared it and don't forget to put

positive and negative

square root of x squared is X

and then copy the positive and negative

and then the square root of 100 this

since this one is a perfect square

that is 10. and as you can see

that's the answer

the solutions of this quadratic equation

are positive

or negative 10. now let's move on with

item number two for number two

we are given this thing

so following this pattern transpose

a to the other side of the equation

we have nine

x squared

is equal to

positive eight

now what will happen here I will adjust

the paper first

so earlier

X or the square root but this time we

can do dividing by nine dividing by nine

cancel cancel

you have your x squared

is equal to

8 over

9. what's next since we already have

this point kind of

format

extract the square root

get the square root

and don't forget the positive and

negative

square root of x squared is

X

well this one

we have here the square root of eight

the square root of eight eight is not a

prefix clear

so to start the square root of eight

square root of 8 is the same as

square root of 4 times 2. the square

root of four is two extract this

will become 2 square root of 2 is square

root of 2. so your square root of 8 will

become 2 square root of 2 so copy the

positive and negative

then for the numerator 2 square root of

2 over

the square root of nine which is three

and as you can see these are the values

of X we have the first solution which is

the positive one positive 2 or 2 square

root of 2 over 3 and the other solution

is negative 2 square root of two over

three okay so let's move on with item

number three suggest this paper

set this one

follow this pattern

okay

transpose this to the other side

we have

2 x squared is equal to positive three

then divide both X by 3

Pi by 2 rather

by 2 cancel

you have your x squared

is equal to

3 over 2.

so as you can see we can extract the

square root already

get the square root guys

don't forget the positive and negative

and I will put it here

square root of x squared is X

then copy positive and negative now

in our fraction three over two we cannot

extract the squared anymore so it will

remain as

positive negative square root of three

over two this one is the final answer

it will be accepted by your teacher

don't worry number four we have x

squared plus 25

here

following this pattern transpose this

one

we have x squared

is equal to negative

25.

now

extract the square root

extract the square

okay

square root of this one is X

and this time

what we have here is this

positive negative now this is negative

guys

is

imaginary is a negative number

automatically it will become imaginary

imaginary number or I

so square root of negative 25 nothing

can be factored out as

square root of

negative 1 times 25 as you can see in 25

knot in the beginning it became positive

after nothing it factor out you negative

one

and remember

now you're adding square root of

negative one

is equal to I by Latin extraction so it

will become like this

so we have

square root of negative one

times the square root of 25.

so since

square root of negative 1 is equal to I

is

and this is

square root of 25 is

times 5 or

5

I

5 I meaning here is X

is positive negative

5 I and this is the correct answer guys

okay

so I hope guys you learned something

from this video and I will create a part

two of this video

examples to

to discuss with about solving quadratic

Creations by extracting the square roots

so guys if you're new to my channel

don't forget to like And subscribe but

hit the Bell button

for you to be updated latest uploads

again it's me teacher gone

bye-bye