Sum and Product of the Roots of Quadratic Equation - Finding the Quadratic Equation

Summary

In 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

00:00

📘 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.

05:02

🔍 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.

10:05

📢 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.

Keywords

💡Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form ax^2 + bx + c = 0. In the video, the main theme revolves around understanding and solving quadratic equations, specifically focusing on finding the sum and product of their roots. The script uses the example of the equation x^2 - 12x + 20 = 0 to demonstrate how to find these roots.

💡Roots

In the context of the video, roots refer to the values of the variable x that satisfy the quadratic equation, making the equation true (i.e., both sides of the equation are equal). The video explains how to find the sum and product of these roots using both manual factoring and the quadratic formula.

💡Sum of the Roots

The sum of the roots of a quadratic equation is calculated using the formula x1 + x2 = -b/a. The video demonstrates this by showing that for the equation x^2 - 12x + 20 = 0, the sum of the roots is 12, which is obtained by adding the roots found through factoring.

💡Product of the Roots

The product of the roots of a quadratic equation is given by the formula x1 * x2 = c/a. The video explains that for the same equation x^2 - 12x + 20 = 0, the product of the roots is 20, which is found by multiplying the roots derived from factoring.

💡Factoring

Factoring is a method of solving quadratic equations by expressing the equation as a product of simpler equations (factors). The video uses factoring to solve x^2 - 12x + 20 = 0, resulting in (x - 10)(x - 2) = 0, which directly gives the roots of the equation.

💡Quadratic Formula

The quadratic formula is a standard algebraic method for finding the roots of a quadratic equation. The formula is x = [-b ± sqrt(b^2 - 4ac)] / (2a). Although not explicitly used in the video, it is related to the formulas for the sum and product of the roots, which are derived from the quadratic formula.

💡Coefficients (a, b, c)

In a quadratic equation ax^2 + bx + c = 0, a, b, and c are the coefficients of the equation. The video emphasizes the importance of these coefficients in the formulas for the sum and product of the roots, where a is the coefficient of x^2, b is the coefficient of x, and c is the constant term.

💡FOIL Method

The FOIL method is a technique used for multiplying two binomials. The video uses this method to expand the factors (x - 10)(x - 2) to derive the original quadratic equation, demonstrating how the roots can be used to reconstruct the equation.

💡Cross Multiplication

Cross multiplication is a technique used to solve equations involving fractions. In the video, it is used to convert the roots given as fractions into a form that can be factored, such as converting 1/3 and 6 into 3x - 1 and x - 6, respectively.

💡Transposing

Transposing in algebra involves moving a term from one side of an equation to the other, often changing its sign in the process. The video demonstrates transposing when it converts the roots into factors of the quadratic equation, such as changing x = -10 to x + 10 = 0.

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