Sum and Product of the Roots of Quadratic Equation - Finding the Quadratic Equation
Summary
TLDRIn this video, the host explains how to find the sum and product of the roots of a quadratic equation. Using the example x² - 12x + 20 = 0, they first solve it manually by factoring, and then apply the formulas: sum of the roots is -B/A and product is C/A. They also demonstrate how to derive the quadratic equation from given roots. The host further illustrates these steps with detailed explanations and encourages viewers to engage with the channel for more math tutorials.
Takeaways
- 📘 The video discusses the sum and product of the roots of a quadratic equation.
- 🔢 The standard form of a quadratic equation is ax^2 + bx + c = 0.
- ➕ The sum of the roots x_1 + x_2 is given by -b/a.
- ✖️ The product of the roots x_1 × x_2 is given by c/a.
- 📚 The video demonstrates solving for the sum and product of the roots of the equation x^2 - 12x + 20 = 0 by factoring.
- 🔍 By factoring, the roots are found to be x = 10 and x = 2, leading to a sum of 12 and a product of 20.
- 📐 The video shows how to use the formulas for the sum and product of the roots without factoring, using the coefficients from the equation.
- 🔄 The process is reversed to find the original quadratic equation given the roots, using the method of forming factors from the roots.
- 📝 The video provides an example of constructing a quadratic equation from given roots (-5, 4) and (1/3, 6) using the foil method.
- 💡 The video concludes with a call to action for viewers to like, subscribe, and turn on notifications for the channel.
Q & A
What is the standard form of a quadratic equation?
-The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a \neq 0 \).
What is the formula for finding the sum of the roots of a quadratic equation?
-The sum of the roots of a quadratic equation is given by the formula \( x_1 + x_2 = -\frac{b}{a} \).
How do you calculate the product of the roots of a quadratic equation?
-The product of the roots of a quadratic equation is calculated using the formula \( x_1 \times x_2 = \frac{c}{a} \).
In the given script, what is the quadratic equation that is being solved?
-The quadratic equation being solved in the script is \( x^2 - 12x + 20 = 0 \).
What are the roots of the equation \( x^2 - 12x + 20 = 0 \) as found in the script?
-The roots of the equation \( x^2 - 12x + 20 = 0 \) are \( x = 10 \) and \( x = 2 \).
How does the script demonstrate the sum of the roots for the equation \( x^2 - 12x + 20 = 0 \)?
-The script demonstrates the sum of the roots by adding the roots \( x = 10 \) and \( x = 2 \), resulting in a sum of 12.
What is the product of the roots for the equation \( x^2 - 12x + 20 = 0 \) as shown in the script?
-The product of the roots for the equation \( x^2 - 12x + 20 = 0 \) is calculated as \( 10 \times 2 = 20 \).
How does the script use the formula to find the sum of the roots of the equation \( x^2 - 12x + 20 = 0 \)?
-The script uses the formula \( x_1 + x_2 = -\frac{b}{a} \) with \( a = 1 \) and \( b = -12 \), resulting in a sum of 12.
What is the product of the roots found using the formula in the script for the equation \( x^2 - 12x + 20 = 0 \)?
-Using the formula \( x_1 \times x_2 = \frac{c}{a} \) with \( c = 20 \) and \( a = 1 \), the product of the roots is found to be 20.
How does the script explain finding the original quadratic equation given the roots?
-The script explains that if given roots, you can form factors based on those roots and then use the FOIL method to expand these factors into the original quadratic equation.
What is the quadratic equation formed with roots -5 and 4 as demonstrated in the script?
-The quadratic equation formed with roots -5 and 4 is \( x^2 + x - 20 = 0 \), derived by factoring \( (x + 5)(x - 4) = 0 \).
How does the script handle the case where the roots are \( \frac{1}{3} \) and 6?
-The script handles this by cross-multiplying to get \( 3x - 1 \) and \( x - 6 \), then using the FOIL method to expand these factors into the quadratic equation \( 3x^2 - 19x + 6 = 0 \).
Outlines
📘 Sum and Product of Quadratic Roots
The script introduces the concept of the sum and product of the roots of a quadratic equation. It explains that for a quadratic equation in the standard form ax^2 + bx + c = 0, the sum of the roots (x1 + x2) is given by -b/a and the product of the roots (x1 * x2) is c/a. The script then demonstrates how to find these values both by factoring a given quadratic equation (x^2 - 12x + 20 = 0) and by applying the formulas. The example shows that the roots are 10 and 2, leading to a sum of 12 and a product of 20. The explanation also covers how to use the formulas when a, b, and c values are known.
🔍 Constructing Quadratic Equations from Roots
This section of the script teaches how to construct a quadratic equation when the roots are known. It provides two examples: one with roots -5 and 4, and another with roots 1/3 and 6. For the first example, the script shows how to form the factors (x + 5) and (x - 4) and then uses the FOIL method to expand them into the quadratic equation x^2 + x - 20 = 0. For the second example, it demonstrates cross-multiplication to form the factors (3x - 1) and (x - 6), and then uses the FOIL method to expand them into the quadratic equation 3x^2 - 19x + 6 = 0. The script emphasizes the process of reversing from the roots to construct the original quadratic equation.
📢 Closing Remarks and Call to Action
The final paragraph serves as a closing to the video script. The speaker, Trigon, encourages viewers to like, subscribe, and hit the bell button for updates on the latest uploads. It ends with a friendly farewell from Trigon.
Mindmap
Keywords
💡Quadratic Equation
💡Roots
💡Sum of the Roots
💡Product of the Roots
💡Factoring
💡Quadratic Formula
💡Coefficients (a, b, c)
💡FOIL Method
💡Cross Multiplication
💡Transposing
Highlights
Introduction to the sum and product of the roots of a quadratic equation.
Explanation of the standard form of a quadratic equation.
Presentation of the formula for the sum of the roots: x1 + x2 = -B/A.
Presentation of the formula for the product of the roots: x1 · x2 = C/A.
Example problem given: find the sum and product of the roots of x^2 - 12x + 20 = 0.
Manual solving of the example problem by factoring.
Identification of the roots as 10 and 2 through factoring.
Calculation of the sum of the roots: 10 + 2 = 12.
Calculation of the product of the roots: 10 × 2 = 20.
Demonstration of using the formula for the sum of the roots with the values A, B, and C.
Demonstration of using the formula for the product of the roots with the values A, B, and C.
Verification that the formula yields the same results as manual solving.
Introduction to finding the original quadratic equation given the roots.
Methodology for constructing the quadratic equation from given roots.
Example of constructing the quadratic equation from roots -5 and 4.
Example of constructing the quadratic equation from roots 1/3 and 6.
Explanation of the FOIL method for combining binomials to form a quadratic equation.
Final quadratic equations obtained from the given roots.
Closing remarks and call to action for viewers to like, subscribe, and enable notifications.
Transcripts
hi guys it's me the Trigon in today's
video we will talk about the sum and
product of the roots of quadratic
equation
now this is the standard form of any
quadratic equation and the formula in
getting the sum and the product of the
roots is simply
x sub 1 plus x sub 2 that represents
your
roots
is equal to negative B over a
while the product is x sub 1 times x sub
2 is equal to C over a so without
further Ado let's do this topic so we
have here these problems or this problem
it says here find the sum and product of
the roots of the equation x square minus
12x plus 12 is equal to 0. so what does
what is what does it mean person I've
been adding some product meaning we need
to add the roots of the equation
and the product
and the roots of and we need to multiply
and add the roots of the quadratic
equation so let's say for example we
will try to solve this manually and
after that we will try to use the
formula
if we have x squared
minus 12x
plus 20 is equal to zero by factoring we
can solve this problem
the factors are
X
minus 10
and x minus 2 because if we have
negative 10 times negative 2 that is
positive 20
. if we have negative 10 plus negative 2
that is negative 12. so solving this by
factoring we have x minus 10
is it called zero
and the other is x minus 2 is equal to
zero transpose this to the other side
your X is equal to positive 10. this is
the first truth or the value of x sub y
well for x sub 2 we have x minus 2 is
equal to zero
transpose this to the other side of the
equation
that is X is equal to positive two or
two now
let us try
and find the sum of the roots to find
the sum of the roots we need to add the
x sub 1 and x sub 2. we have x sub 1
plus x sub 2
and we have 10
plus 2 meaning
the sum is equal to
12. and as for the products product of
the roots we have the x sub 1
times the x sub 2
and that is simply
10
times 2. and that will give you the
answer of
20. now
this method is by manually
solving for the value of x and
eventually adding multiplying
the roots to get the sum and product
so for the next method we will use this
formula okay
the formula
in getting the sum
x sub 1 plus x sub 2
is simply
negative B over
a
now
for the product command we have x sub 1
times x sub 2
in the formula simply
C over a and as you can see
in this formulas we need to identify the
values of a b and c in this problem
your a
is equal to one
your B is definitely negative 12. and
your C is
20.
now let us use this formula and
substitute this values of a b and c
for the sum we have negative
original negative
and then your B is negative 12 so we
will use parenthesis
to indicate multiplication over
here a which is equal to 1. so
simplifying this
negative times negative is positive so
this is 12
over 1 and simplifying 12 over 1 that
will give the answer of
12. and as you can see
using the formula
and by manually adding the roots of
the given quadratic equation
we can still have the same answer
okay so much better to use this formula
next
to find a product
use the value of C which is 20.
over your a which is equal to 1. and
simplifying this 20 divided by 1 is
simply
20. as you can see we still have the
same answer so I hope guys
on how to find the solving product of
the roots of the equation
and another possible problems that you
will encounter is that you will be asked
to find the original equation if given
ammonium roots
okay
let's try these problems
find equation or the quadratic creation
given the following Roots so here number
one let me explain
negative five and four are the roots of
quadratic equation so in other words
it is simply like this if these are the
roots we have X is equal to negative
five
and X
is equal to 4.
now the question here if we have this x
is equal to negative 5 and X is equal to
negative 4. how do we find the original
quadratic equation
given this Roots so what we need to do
here is we will reverse the process so
here
if we have x equal to negative 5 we will
transpose back negative five
from right to left
so we will put it here from right to
left and here from right to left it will
become
X plus five is equal to zero
and the other is
X
minus four
is equal to zero so we know that these
are the factors we will Express this as
factors and that will be equal to zero
so we have X
plus five
times x minus four and all of them are
equal to zero
in this case
we will use the foil method to finally
get the quadratic equation we have x
times x which is x squared
x times negative 4 that is negative 4X
then 5 times x we have plus 5X
negative 5 times negative four
negative 20 or minus 20 is equal to zero
and
as you can see we can still combine
negative four and five x so we have x
squared
and this is plus X and for this minus 20
is equal to zero and
this is the original quadratic equation
what about this number two
number two we have one third and six
here the first is X
is equal to one-third
this x is equal to
six
for this one
example or for this value
we need to do cross multiplication first
okay we will cross multiply or we will
multiply
this by three by three
three times x is 3x
here
1 times 3 is 3 divided by three
that is equal to
1.
and this part transpose 6 to the other
side from positive it will become
negative
x minus 6 is equal to zero foreign
this positive one must be transposed to
the other side it will become 3x
-1 is equal to zero and this is x minus
six
is equal to zero
we will Express them in as factors and
then b equal to zero this 3x
-1
times x
minus six
is equal to zero
same thing we need to do the foil method
3 times x that is 3
x square
3x times negative 6 that is negative 18
x
followed by negative 1 times x
that is negative X
negative 1 times negative 6 that is
positive 6 or 6. is equal to zero
combine this copy 3x square
okay copy 3x squared
this is negative 19 x because we have
here an invisible one our invisible
negative one
then plus six
is equal to zero and this is now
the quadratic equation of
the given Roots one third and six so if
you're new to my channel don't forget to
like And subscribe but hit the Bell
button for you to be updated setting
latest uploads again it's meet turgon
bye-bye
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