How to Solve Quadratic Equations by Extracting the Square Root? @MathTeacherGon

Summary

In this video, the teacher explains how to solve quadratic equations using the method of extracting square roots. This is one of several methods for solving quadratic equations, alongside factoring, completing the square, and using the quadratic formula. The video demonstrates the process with multiple examples, highlighting how to handle different forms of equations by isolating terms and extracting square roots to find both positive and negative solutions. The teacher emphasizes the importance of understanding perfect squares and offers a step-by-step approach to manipulate equations, making the process easier to grasp.

Takeaways

• 📚 The video focuses on solving quadratic equations by extracting square roots.
• 🔍 Extracting the square root is one of several methods to solve quadratic equations, alongside factoring, completing the square, and the quadratic formula.
• 🧮 If x² = k, then x = ±√k for all non-negative real numbers k.
• ✔️ When extracting the square root of a number, remember that there are always two solutions: one positive and one negative.
• 🔢 Example 1: x² = 49 gives solutions of x = ±7.
• ➕ To solve equations not in the standard form, manipulate the equation (e.g., adding or subtracting terms to isolate x²).
• 🔑 Example 2: x² - 8 = 1 is manipulated to x² = 9, yielding x = ±3.
• 🔧 For non-perfect squares (like √27), factor the number into a perfect square and simplify.
• 🌀 More complex quadratic equations, like (x - 2)² = 16, can be solved by extracting the square root and further isolating x.
• 📝 The instructor assigns item 7 as homework for viewers to practice solving quadratic equations.

Q & A

What is the focus of the video?

The video focuses on solving quadratic equations by extracting square roots, one of the methods used to solve quadratic equations.

What other methods for solving quadratic equations are mentioned in the video?

The methods mentioned are factoring, completing the square, and using the quadratic formula.

What property is essential when extracting square roots to solve quadratic equations?

If x^2 = k, then x = ±√k, meaning that the square root of a number gives two values: one positive and one negative.

What is the result of solving x^2 = 49 by extracting square roots?

The solution is x = ±7, meaning x = 7 and x = -7.

How do you manipulate an equation like x^2 - 8 = 1 to solve by square roots?

First, add 8 to both sides to isolate x^2, resulting in x^2 = 9, then take the square root to get x = ±3.

What do you do if the number under the square root is not a perfect square, like x^2 = 27?

You can factor the number into perfect squares and simplify. In this case, x = ±3√3.

How do you solve an equation like (x - 2)^2 = 16 using square roots?

Take the square root of both sides to get x - 2 = ±4, then add 2 to both sides to find x = 6 and x = -2.

What is the first step in solving an equation like 2x^2 - 18 = 0?

Add 18 to both sides to get 2x^2 = 18, then divide both sides by 2 to simplify the equation.

How do you handle an equation like (x + 10)^2 = 25?

Take the square root of both sides to get x + 10 = ±5, then subtract 10 from both sides to find the solutions x = -5 and x = -15.

What are the possible values of x if k is positive, zero, or negative in a quadratic equation?

If k is positive, there are two real values for x. If k is zero, there is one real value. If k is negative, there are no real values for x.

Outlines

00:00

📚 Introduction to Solving Quadratic Equations by Extracting Square Roots

The video begins with a teacher introducing the topic of solving quadratic equations using the square root extraction method. This is one of four key methods (factoring, square roots, completing the square, and using the quadratic formula). The teacher recaps that in the previous video, factoring was discussed. A key principle is outlined: if x² = k, then x equals positive or negative square root of k. The importance of recognizing non-negative real numbers and understanding the properties of square roots is emphasized. The teacher explains the concept by solving examples and reiterating that quadratic equations typically have two solutions.

05:01

🧮 Example 1: Solving x² = 49

The teacher walks through solving the equation x² = 49, explaining that it fits the x² = k pattern. They demonstrate extracting the square roots from both sides of the equation, resulting in x = ±7, meaning the two possible solutions are x = 7 and x = -7. The teacher also reminds students to memorize the square roots of perfect squares to make solving such equations faster.

10:01

🔢 Example 2: Solving x² - 8 = 1

In the second example, the equation x² - 8 = 1 is not in the correct form. The teacher explains how to manipulate the equation by adding 8 to both sides, turning it into x² = 9. Extracting the square roots of both sides results in x = ±3, meaning the two possible solutions are x = 3 and x = -3.

📝 Example 3: Solving x² + 4 = 31

The third example follows a similar process to example 2. The equation x² + 4 = 31 is manipulated by subtracting 4 from both sides to isolate x², leading to x² = 27. Since 27 is not a perfect square, the teacher explains how to simplify the square root by factoring it as √(9 * 3), resulting in x = ±3√3.

📐 Example 4: Solving (x - 2)² = 16

In this example, the equation involves a binomial square. The teacher shows how to extract the square root of both sides, leading to x - 2 = ±4. To isolate x, they add 2 to both sides, resulting in two solutions: x = 6 and x = -2.

🧠 Example 5: Solving 2x² - 18 = 0

For example 5, the equation has a coefficient of 2 in front of x². The teacher first adds 18 to both sides to isolate the quadratic term, resulting in 2x² = 18. They then divide by 2 to get x² = 9 and extract the square root, giving x = ±3.

📝 Example 6: Solving (x + 10)² = 25

In this example, the teacher demonstrates solving the binomial square (x + 10)² = 25 by extracting the square root from both sides. This results in x + 10 = ±5. The teacher then isolates x by subtracting 10 from both sides, giving two solutions: x = -5 and x = -15.

🎯 Final Notes and Assignment

The teacher summarizes the method of solving quadratic equations by extracting square roots, highlighting that the equation should first be manipulated into the form x² = k before extracting roots. The teacher explains that if k is positive, two solutions exist; if k is zero, there is only one solution; and if k is negative, no real solution exists. Students are assigned a final problem to solve using this method and are encouraged to follow the teacher on various social media platforms for more content.

Keywords

💡Quadratic Equations

Quadratic equations are mathematical expressions where the highest exponent of the variable is 2, typically in the form ax^2 + bx + c = 0. In the video, the teacher focuses on methods to solve these equations, specifically through extracting square roots, which is one of the four main methods discussed.

💡Extracting Square Roots

Extracting square roots is a method used to solve quadratic equations when the equation is in the form x^2 = k. The process involves taking the square root of both sides, resulting in two possible values: one positive and one negative. This method is emphasized in the video as a direct way to find the roots when no other terms are present.

💡Perfect Squares

Perfect squares are numbers that are the product of an integer multiplied by itself, such as 1, 4, 9, 16, etc. In the context of the video, knowing perfect squares is crucial for quickly solving quadratic equations by extracting square roots, as it allows the student to immediately identify the roots without further calculations.

💡Factoring

Factoring is another method of solving quadratic equations where the equation is rewritten as a product of two binomials set to zero. The video briefly mentions factoring as one of the four key methods for solving quadratic equations, providing context that extracting square roots is an alternative approach.

💡Completing the Square

Completing the square is a technique to solve quadratic equations by making one side of the equation a perfect square trinomial. This method is one of the four methods mentioned in the video, though the focus remains on extracting square roots. Understanding this method helps illustrate the different ways to manipulate equations.

💡Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation, given as x = (-b ± √(b^2 - 4ac)) / 2a. While the teacher mentions this formula as part of the four methods, the focus of the video is on simpler cases where extracting square roots can be applied directly.

💡Roots of Equations

The roots of a quadratic equation are the solutions or values of x that satisfy the equation. The video repeatedly emphasizes finding both positive and negative roots, illustrating how each quadratic equation typically has two roots, unless it results in a single root (when k = 0) or no real roots (when k is negative).

💡Manipulating Equations

Manipulating equations involves rearranging terms to isolate the variable squared term on one side of the equation. The video demonstrates this process in several examples, where constants are moved across the equals sign to set up the equation for square root extraction, which is a crucial step for this solving method.

💡Square Root Property

The square root property states that if x^2 = k, then x = ±√k, applicable for all non-negative real numbers k. This principle is the foundation of the video’s lesson, guiding the approach to solve quadratic equations using square roots and helping students understand why each solution yields two values.

💡Non-Negative Real Numbers

Non-negative real numbers are numbers that are either positive or zero, excluding negative values. In the context of extracting square roots, the teacher emphasizes that the square root property only applies when k is non-negative, as negative values would result in complex roots, which are not covered in this video.

Highlights

Introduction to solving quadratic equations by extracting square roots, a method alongside factoring, completing the square, and using the quadratic formula.

Key property: If x² = k, then x = ±√k for non-negative real numbers k.

Example 1: Solving x² = 49 by extracting the square root, giving solutions x = ±7.

Emphasis on memorizing perfect squares to make solving quadratic equations faster.

Example 2: Solving x² - 8 = 1 by adding 8 to both sides, then extracting square roots to find x = ±3.

Example 3: Solving x² + 4 = 31 by subtracting 4 from both sides and extracting square roots, resulting in x = ±3√3.

Example 4: Solving (x - 2)² = 16 by extracting square roots and isolating x, leading to solutions x = 6 and x = -2.

Explaining the importance of transposing terms and isolating x during solving quadratic equations.

Example 5: Solving 2x² - 18 = 0 by transposing and dividing, then extracting square roots to get x = ±3.

Reminder to viewers: Follow on YouTube for more math videos and subscribe for regular content.

Example 6: Solving (x + 10)² = 25 by extracting square roots and transposing, yielding x = -5 and x = -15.

Summarizing the method of extracting square roots, emphasizing how to manipulate equations to isolate x² and extract roots.

Clarifying that if k is positive, two values for x are found; if k is zero, only one value exists; if k is negative, no real solution exists.

Assignment for students: Solve a given quadratic equation using the method of extracting square roots.

Outro: Encouraging viewers to follow on social media and reminding them of the value of consistent practice in math.

Transcripts

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

video we will talk about solving

quadratic equations

by extracting the square roots

last night we have uploaded a video with

regard to solving quadratic equations

in by factoring and you can see it here

and also i can leave the

link satin description box

so without further ado

let's do this topic

so before we start solving quadratic

equations

by extracting the square roots let me

remind you that

this method extracting the square root

is one of the methods

in solving quadratic equations so we

have

factoring

extracting the square roots

completing the square and using the

quadratic formula

and

sign i'm a master knowing topic nothing

about this so before we start

melon tylenol square the property in

solving quadratic equations cyberito

if x squared is equal to k

then

x is equal to positive negative square

root of k

for all

non-negative real numbers k

so little

when you extract the square root of x

squared that is equal to x

and

when we extract

an negative real number k

it will give us

two different values

one is positive and the other is

negative

so for you to find out more about this

property and about this

method let us solve different examples

of quadratic equations

in number one we have x squared

is equal to 49 as you can see

this one is already in the form x

squared is equal to k so basically

what we need here

is to extract the square root of

each of the side of the equations like

this one

let's get the square root of x squared

and the square root of

49.

so again guys um another reminder of

allah

if you want to know or if you want na

mas madaliya maging solving morito you

need to memorize the perfect squares and

the square root of perfect square

numbers okay let's continue

the square root of x squared is x

and based

on our given squared property

upon the extractor and square root

it will give us two different

values one is positive and the other is

negative so the square root of 49 is not

just seven

that is

positive negative

seven

so

what does it mean guys if we have x is

equal to

positive seven and negative seven it

simply means that your

the first the first value of x

or x sub one is

simply positive seven or seven

and the second value or the second root

of this equation is negative

seven that's it guys

now let's continue with item number two

in number two we are given x squared

minus eight is equal to one so here

it's not yet

in this kind of a pattern or formation

so what we need to do is to manipulate

the equation

so we need to eliminate negative eight

here to make it

here on the other side of the equation

so

what's the step

to eliminate this all we need to do is

to add eight

on both sides of the equation

so as you can see this negative eight

plus eight that is zero so it will be

eliminated so what will remain here

is x squared

and on the other side of the equation we

have one plus eight

and that is equal to nine and as you can

see

we have the same pattern

as number one and as this

okay

so next step not in detail if we reach

this kind of pattern

is that we will get the square root

of this equation

okay remember we have positive and

negative

the square root of x squared is x

and then the square root of 9 is simply

positive negative 3 again so what does

it mean

you have two different routes you have

positive three and negative three

so let's continue with item number three

and by the way guys

uh for those who are watching from

tiktok

let me remind you that we have our

youtube channel which is mattychergon

you can subscribe here to follow and

watch more videos about your grade

levels so let's continue with number

three

for number three we have x squared plus

four

is equal to thirty one

same pattern with item number two

we need to manipulate the equation

in other words we need to eliminate plus

four so what we need to do

is is to subtract four

both sides of the equation

so this will become zero okay or this

will become eliminated

so we will copy x squared

and then

31 minus 4

is

what

20

7

okay so for the 27

we need to get the square root of

this

and this and remember meron count

positive negative

pero

27 is not a perfect square so how are we

going to simplify this so remember

we can

factor out 27

like this one

nine times three that is twenty-seven

the square root of nine is three

so what we have here

is a square root of 3 meaning

to simplify

square root of 27 since this one is not

a perfect square you need to factor it

out

so one must be a perfect square and the

other is not so it will become three

square root of three and the answer here

is simply

we will continue it here

is x is equal to positive negative three

square root of three

okay so these are the possible roots

so the roots are

three square root of three

or negative three square root of three

so let's continue with number four

in number four this one is quite

different from the previous examples as

you can see we have here x minus 2

squared

is equal to 16. don't worry because this

one is not difficult

so as you can see this one is already in

this pattern

so we can easily extract the square

roots

here

get the square root of this and remember

you have positive and negative

in getting the square root of this all

you need to do is to cancel out this one

and your

exponent

so what will remain is that we only have

x

minus 2

and on the other side of the equation

the square root of 16

is equal to

positive negative 4

but we're not yet done because this one

is not

the variable x is not yet isolated so

what we need to do

is to eliminate negative 2 by adding

both sides of the equation by

2. so what will happen

so

instead of adding i will just transpose

this to the other side of the equation

so it will become x is equal to positive

negative four

from negative when you transpose a term

it will become

positive two

so we will solve for

the first value of x x sub one

so how

use the first positive four

use positive four that is

four

plus two

meaning

your x sub 1 is equal to

6 this is the first

root or solution of this quadratic

equation

next

for x sub 2

since we are done using the positive

four again

in x sub one we use positive four in x

sub two we will use the negative four

so that's negative four

plus two

and x sub two

is simply negative two

and these are the roots of

the fourth quadratic equation

six and negative two

let's continue with more examples

we have here number five six and seven

in this number seven this will serve as

your assignment okay

for number seven

we have here two x squared minus

eighteen is equal to

zero first

as you can see your x squared has the

coefficient of two so our target here is

to eliminate that coefficient

so i will trans transpose first 18 or i

will add both sides of the equation by

18.

so plus 18

so it will become this one will become

zero so we have now

two x squared

is equal to zero plus eighteen which is

equal to eighteen

our next step

is to cancel out two by dividing both

sides of the equation by 2.

cancel cancel you have now

x squared

is equal to 18 divided by 2 which is

equal to

9.

extract the square root

extract the square root don't forget the

positive and negative

so we have here

x

is equal to the square root of nine

which is positive negative three and

this is the answer here

okay

next let's move on to item number six

for number six

um

this one needs a little bit of

manipulation

we need to eliminate first 25

by adding

25 on both sides of the equation

this will become zero so it will remain

here on the left side of the equation is

simply

x

plus 10

squared

and on the other side we have zero plus

25

so that is 25

so as you can see

this pattern or this kind of equation is

the same as

this in item number four

so what we need to do

is to extract the square root

get the square root

get the square root and this is positive

negative so we need to eliminate this

and this exponent so what we have now is

x

plus 10

is equal to

positive negative square root of 25

which is equal to 5.

our next step

is transpose 10 to the other side it

will become x is equal to positive

negative 5

from positive it will become negative

10.

so now

we are ready to compute for x sub 1.

your x sub 1

is first using the positive 5 that is 5

minus 10

meaning your x sub 1 is simply negative

5.

this is the first solution

now

for x sub 2 or the second solution

we have to use

the negative 5

we have negative 5

minus 10

so we have x sub 2

is equal to negative 15

and voila

this is the second value of x

so let's summarize first guys

so what we need to do here in extracting

the square roots is that we need to

convert or manipulate the equation

in which

our target is to

to copy this kind of form we have x

squared support okay and then extract

the square roots and remember

if your

k is positive remember guys if your k is

positive

you can have two different values of x

okay

and if your k is zero you only have one

value of x and if your k is negative

1 real value of x

and

for number 7 this will serve as your

assignment

okay

so guys

if you're new to my channel don't forget

to like and subscribe and also

you can follow me here

starting facebook page

we have

this facebook page ako citychargon

follow me here guys

and

bye