Solving Quadratic Equations using Quadratic Formula

Summary

In this educational video, the host, Caregon, introduces viewers to solving quadratic equations using the quadratic formula. The script walks through the process of solving the equation 2x^2 - 4x - 30 = 0 step by step, emphasizing the importance of expressing the equation in standard form to identify coefficients a, b, and c. The video demonstrates the formula x = (-b ± sqrt(b^2 - 4ac)) / 2a and simplifies it to find the roots, which are x = 5 and x = -3. The host encourages viewers to learn and apply this method, promoting engagement with the content.

Takeaways

• 📚 The video explains how to solve quadratic equations using the quadratic formula.
• ✏️ The quadratic formula is used as an alternative method to factoring, completing the square, and square roots.
• 🧮 The formula for solving quadratic equations is X = (-B ± √(B² - 4AC)) / 2A.
• 🔍 Before applying the formula, the equation must be in standard form, where A, B, and C are identified.
• ➗ In the example, 2x² - 4x - 30 = 0, the values of A, B, and C are 2, -4, and -30 respectively.
• 🔢 The discriminant (B² - 4AC) is calculated as 16 + 240, giving a total of 256.
• 🟢 Since the square root of 256 is 16, the equation becomes X = (4 ± 16) / 4.
• ➕ The first solution (X₁) is calculated as (4 + 16) / 4 = 5.
• ➖ The second solution (X₂) is calculated as (4 - 16) / 4 = -3.
• ✅ The final solutions for the quadratic equation are X = 5 and X = -3.

Q & A

What is the main topic of the video?

The main topic of the video is solving quadratic equations using the quadratic formula.

What are the alternative methods mentioned for solving quadratic equations?

The alternative methods mentioned are factoring, completing the square, and using square roots.

What is the quadratic formula used to solve quadratic equations?

The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a.

What is the first step in using the quadratic formula?

The first step is to express the equation in standard form to identify the values of a, b, and c.

What are the values of a, b, and c in the example equation 2x^2 - 4x - 30 = 0?

In the example equation, a is 2, b is -4, and c is -30.

How do you calculate the discriminant in the quadratic formula?

The discriminant is calculated as b^2 - 4ac.

What is the discriminant for the example equation 2x^2 - 4x - 30 = 0?

The discriminant is (-4)^2 - 4(2)(-30) = 16 + 240 = 256.

What are the two solutions for the example equation using the quadratic formula?

The two solutions are x = 5 and x = -3.

What does the plus-minus symbol in the quadratic formula represent?

The plus-minus symbol indicates that there are two possible solutions, one using addition and the other using subtraction.

How does the video encourage viewers to engage with the content?

The video encourages viewers to like and subscribe for updates on the latest uploads.

Outlines

00:00

📘 Introduction to Solving Quadratic Equations

This paragraph introduces the topic of solving quadratic equations using the quadratic formula. The speaker, identified as 'the caregon,' presents this method as an alternative to other common techniques such as square roots, completing the square, and factoring. The focus is on the quadratic formula, which is used to solve the equation 2x^2 - 4x - 30 = 0. The speaker explains the process of expressing the equation in standard form to identify the coefficients a, b, and c, which are crucial for applying the formula. The values for a, b, and c are identified as 2, -4, and -30, respectively. The formula is then applied step by step, with the speaker detailing each calculation, including the simplification of the square root and the final solution for x.

05:01

🔢 Solving the Quadratic Equation Step by Step

In this paragraph, the speaker continues the explanation of solving the quadratic equation 2x^2 - 4x - 30 = 0 using the quadratic formula. The process involves calculating the discriminant (B^2 - 4AC) and then solving for x by taking the square root of the discriminant and dividing by 2a. The speaker simplifies the discriminant to 256 and then takes the square root to find the values of x. Two solutions are presented: x = 5 and x = -3. The speaker emphasizes the importance of understanding each step in the process to successfully solve quadratic equations. The paragraph concludes with a call to action for viewers to like and subscribe to the channel for more educational content.

Keywords

💡Quadratic Equations

Quadratic equations are polynomial equations of degree two, meaning they have the highest power of 2 for the variable. They are commonly expressed in the form ax^2 + bx + c = 0, where a, b, and c are constants. In the video, the speaker focuses on solving quadratic equations using the quadratic formula, which is a method to find the roots of such equations. The example given in the script is 2x^2 - 4x - 30 = 0, which is a typical quadratic equation.

💡Quadratic Formula

The quadratic formula is a standard algebraic method for solving quadratic equations. It is expressed as x = [-b ± sqrt(b^2 - 4ac)] / (2a). This formula allows for finding the roots of any quadratic equation, regardless of whether they are real or complex. The video script demonstrates the application of this formula to the equation 2x^2 - 4x - 30 = 0, showing each step of the calculation to derive the values of x.

💡Standard Form

Standard form in the context of quadratic equations refers to the arrangement of the equation in the form ax^2 + bx + c = 0, with the terms ordered by descending powers of the variable x. The video emphasizes the importance of expressing the equation in standard form before applying the quadratic formula, as it helps in identifying the coefficients a, b, and c, which are essential for the formula.

💡Coefficients

In the context of the video, coefficients refer to the numerical factors of the terms in a quadratic equation. Specifically, 'a' is the coefficient of x^2, 'b' is the coefficient of x, and 'c' is the constant term. The speaker identifies these coefficients from the given equation (a = 2, b = -4, c = -30) and uses them in the quadratic formula to find the solutions.

💡Solving

Solving, in algebra, means finding the values of the variable that make the equation true. In the video, the process of solving involves applying the quadratic formula to the equation 2x^2 - 4x - 30 = 0 to find the values of x. The script outlines each step, from identifying coefficients to simplifying the radical and calculating the roots.

💡Roots

Roots of a quadratic equation are the values of the variable that satisfy the equation, making it true. The video script demonstrates finding the roots of the equation 2x^2 - 4x - 30 = 0 by using the quadratic formula. The roots are the solutions to the equation, and in this case, the video finds two roots: x = 5 and x = -3.

💡Square Root

A square root is a value that, when multiplied by itself, gives the original number. In the quadratic formula, the square root is taken of the discriminant (b^2 - 4ac). The video script shows the calculation of the square root in the context of the quadratic formula, where the discriminant is 16 + 240, leading to the square root of 256, which simplifies to 16.

💡Discriminant

The discriminant of a quadratic equation, denoted as b^2 - 4ac, determines the nature of the roots. If it's positive, the equation has two distinct real roots; if zero, one real root (or two identical real roots); and if negative, two complex roots. In the video, the discriminant is calculated as part of the quadratic formula application, leading to the determination of real roots.

💡Simplification

Simplification in mathematics refers to making an expression easier to understand or calculate by reducing it to its simplest form. The video script includes simplification of the quadratic formula's result, where the radical part and the coefficients are simplified to find the roots of the equation. This process is crucial for obtaining the final solutions for x.

💡Like and Subscribe

These terms are common in the context of video content platforms like YouTube, where creators encourage viewers to 'like' their videos and 'subscribe' to their channels for updates on new content. In the video script, the speaker ends with a call to action for viewers to like and subscribe, which is a standard practice to grow an audience and increase engagement.

Highlights

Introduction to solving quadratic equations using the quadratic formula.

Alternative methods for solving quadratic equations include factoring, completing the square, and using square roots.

Quadratic formula is presented as a method to solve quadratic equations.

Equation 2x^2 - 4x - 30 = 0 is used as an example to demonstrate the quadratic formula.

Explanation of expressing the equation in standard form to identify coefficients a, b, and c.

Identification of the values of a (2), b (-4), and c (-30) for the given equation.

Step-by-step application of the quadratic formula to find the values of x.

Calculation of the discriminant (b^2 - 4ac) and its significance in the quadratic formula.

Simplification of the square root term in the quadratic formula.

Derivation of the two possible solutions for x from the quadratic formula.

Solution x1 = 5 is obtained from the positive square root.

Solution x2 = -3 is obtained from the negative square root.

Final solutions to the equation are x = 5 and x = -3.

Encouragement for viewers to learn from the video and apply the quadratic formula.

Call to action for viewers to like and subscribe for updates on latest uploads.

Conclusion and farewell from the presenter.

Transcripts

hi guys it's me the caregon in today's

video we will talk about solving

quadratic Creations using quadratic

formula

so without further ado

let's do this topic

this topic is an alternative method on

how to solve quadratic equations we have

destruction the square roots completing

the square and factoring and in capacity

that working migration stay on you can

use the quadratic formula to solve

quadratic creations

now

what we have here is 2x squared minus 4X

minus 30 is equal to zero and we will

solve this using the formula X is equal

to negative B positive negative square

root of B squared minus 4AC over to a

what you need to secure here is first

you need to express your equation in

standard form and then eventually you

can get the values of

a

B and C because

to use this formula in this equation

what is the value of a the value of a is

two

the value of B is not 4 that is negative

4.

the value of C is negative

30. now after identifying or determining

the values of a b and c we can use the

formula we have here x

is equal to negative

your B is negative also so you can

enclose it by parenthesis you have

negative

four

then you have positive negative

then after that square root off

B squared this is negative 4 guys

you have to include it by parenthesis

negative four

raised to the second power

and then extend that into

we have four in your formula minus four

then guys for a and c and 4 you need to

express it as in multiplication your a

is two so four times two

times your C is negative 30.

over

2

times your a is to

radical

so um

in this case you have your ex

is equal to negative times negative is

positive so it will become Sim

4

then you have your positive negative

let's simplify the radical part

negative 4 squared is 16.

then this one negative 4 times 2 times

negative 30.

you have negative four times two

that is negative eight so you have

negative eight then you will multiply

negative eight again this negative eight

it came from the product of negative

four and two negative four times two is

negative eight then you will multiply it

this negative eight by negative 30.

negative times negative is positive so

this is Plus

then you have eight times thirty again

eight times thirty that is two hundred

forty

over

2 times 2 which is equal to 4.

we can we cannot cancel out four Hindi

question

next is we need to simplify this radical

16 plus 240 that is 256. so what we have

now is X

is equal to four

positive negative square root of 256

over

4.

and this square of 256 this number is a

perfect square

we can extract it the square root of 256

is 16.

so what we have here is X

is equal to 4.

positive negative 16 over

4. when you are done simplifying the

radical part

we can now solve for the values of X

let's go here let's use this space

let's start with x sub 1.

for the x sub 1

we will use four

then this is positive negative positive

Muna we have plus 16 over

four

simplify this part you have

4 plus 16 which is 20.

over 4 and 20 divided by 4 that is equal

to

5. and as you can see

this is the first

solution of the given equation

the value of x is 5. now let's go with

the x sub 2 or the second root

x sub 2.

for the x sub 2 you have four and we are

done using the positive 16 or plus 16

minamine is minus

16 over

4. simplify

4 minus 16

you have your

negative 12 over 4.

then divide this

negative 12 divided by 4 is negative

three so as you can see this is now the

second value of x we have X is equal to

negative three

now for this given equation the

solutions are 5 and negative three now

guys

I hope you learned something from this

video on how to use the quadratic

formula in solving quadratic equations

now if you're new to my page or Channel

please don't forget to like And

subscribe button

for you to be updated latest uploads

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