How to Solve Quadratic Equations by Extracting the Square Root? @MathTeacherGon
Summary
TLDRIn 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
📚 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.
🧮 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.
🔢 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.
Mindmap
Keywords
💡Quadratic Equations
💡Extracting Square Roots
💡Perfect Squares
💡Factoring
💡Completing the Square
💡Quadratic Formula
💡Roots of Equations
💡Manipulating Equations
💡Square Root Property
💡Non-Negative Real Numbers
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
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