Solving Quadratic Equations by Extracting the Square Roots by @MathTeacherGon
Summary
TLDRIn 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
📘 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.
🔍 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.
📚 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.
Mindmap
Keywords
💡Quadratic Equations
💡Factoring
💡Extracting Square Roots
💡Difference of Two Squares
💡Standard Form
💡Positive and Negative Roots
💡Transpose
💡Imaginary Numbers
💡Square Roots
💡Perfect Squares
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
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