How To Solve Quadratic Equations Using The Quadratic Formula

Summary

This video tutorial offers a step-by-step guide on solving quadratic equations using the quadratic formula. It begins with an example equation, 2x^2 + 3x - 2 = 0, and demonstrates how to identify coefficients a, b, and c. The formula is then applied to find two possible solutions for x, which are verified by substituting back into the original equation. The process is repeated with a second example, showcasing the versatility of the quadratic formula. The video concludes by reinforcing the method's effectiveness in solving quadratic equations, encouraging viewers to practice and master this fundamental mathematical skill.

Takeaways

• 📚 The video is a tutorial on solving quadratic equations using the quadratic formula.
• 🔍 It begins with an example equation: 2x^2 + 3x - 2 = 0, aiming to find the values of x that satisfy the equation.
• 📝 The quadratic formula introduced is: x = -b ± √(b^2 - 4ac) / (2a).
• 📐 The terms a, b, and c are identified as coefficients in the quadratic equation: a is the coefficient of x^2, b is the coefficient of x, and c is the constant term.
• 🔢 The formula is applied to the example equation, with a = 2, b = 3, and c = -2.
• 🧩 The discriminant (b^2 - 4ac) is calculated, which determines the nature of the roots (real and distinct, real and equal, or complex).
• 📉 The discriminant in the example is found to be positive, indicating two distinct real roots.
• 📈 The roots are calculated to be x = 1/2 and x = -2, demonstrating the use of the plus-minus symbol in the formula.
• 🔄 The video suggests checking the solutions by substituting them back into the original equation.
• 📝 A second example is presented with a = 6, b = -17, and c = 12, to further illustrate the application of the formula.
• 🔑 The roots for the second example are found to be x = 3/2 and x = 4/3, showcasing the simplification of fractions.
• 👍 The video concludes by reinforcing the method for solving quadratic equations using the quadratic formula.

Q & A

What is the main topic of the video?

The main topic of the video is how to solve quadratic equations using the quadratic formula.

What is the quadratic formula?

The quadratic formula is used to solve quadratic equations and is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

What are the coefficients a, b, and c in the context of the quadratic formula?

In the quadratic formula, 'a' is the coefficient of \( x^2 \), 'b' is the coefficient of 'x', and 'c' is the constant term.

What is the first example equation given in the video?

The first example equation given is \( 2x^2 + 3x - 2 = 0 \).

How are the values of a, b, and c determined for the first example equation?

For the first example equation, 'a' is 2, 'b' is 3, and 'c' is -2, based on the standard form of a quadratic equation \( ax^2 + bx + c = 0 \).

What is the discriminant in the quadratic formula?

The discriminant in the quadratic formula is the part under the square root, \( b^2 - 4ac \), and it determines the nature of the roots of the quadratic equation.

How many solutions does the first example equation have?

The first example equation has two solutions, as indicated by the 'plus-minus' symbol in the quadratic formula.

What is the second example equation presented in the video?

The second example equation is not explicitly given in the transcript, but it is described to have 'a' as 6, 'b' as -17, and 'c' as 12.

What is the purpose of the 'plus-minus' symbol in the quadratic formula?

The 'plus-minus' symbol in the quadratic formula indicates that there are two possible solutions for 'x', one by adding the square root result to '-b' and the other by subtracting it.

How does the video demonstrate checking the solution of a quadratic equation?

The video demonstrates checking the solution by plugging the found value of 'x' back into the original equation to see if it satisfies the equation, making the left-hand side equal to zero.

What is the final advice given in the video regarding solving quadratic equations?

The final advice is that once you find the solutions using the quadratic formula, you can check your answers by plugging them back into the original equation.

Outlines

00:00

📚 Solving Quadratic Equations Using the Quadratic Formula

This paragraph introduces the quadratic formula as a method for solving quadratic equations. It begins with an example equation, 2x^2 + 3x - 2 = 0, and describes the quadratic formula: -b ± √(b^2 - 4ac) / 2a. The coefficients a, b, and c are identified from the equation, and the formula is applied step-by-step to solve for x. The calculations show that x can be either 1/2 or -2. Finally, the solution is verified by substituting the values back into the original equation.

05:01

📝 Verifying Solutions and Another Example

This paragraph continues with the verification process for the solutions obtained using the quadratic formula. It checks the solution x = -2 by substituting it back into the original equation, confirming it is correct. A new quadratic equation is then introduced: 6x^2 - 17x + 12 = 0. The values of a, b, and c are identified as 6, -17, and 12, respectively. The quadratic formula is applied, showing detailed steps and calculations to arrive at the solutions x = 3/2 and x = 4/3. The paragraph concludes by simplifying the fractions and presenting the final answers.

Keywords

💡Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically in the form of ax^2 + bx + c = 0, where a, b, and c are constants, and x represents an unknown variable. In the video, quadratic equations are the central topic, with the focus on solving them using a specific formula. The script provides an example of a quadratic equation, 2x^2 + 3x - 2 = 0, to demonstrate the solving process.

💡Quadratic Formula

The quadratic formula is a method used to solve quadratic equations. It is given by the expression x = (-b ± √(b^2 - 4ac)) / (2a). The video script explains the steps to identify the coefficients a, b, and c from a given quadratic equation and then apply the formula to find the values of x. This formula is the main tool discussed in the video for solving the presented equations.

💡Coefficients

In the context of the video, coefficients refer to the numerical values that are multiplied by the variable 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 script clarifies how to identify these coefficients from the equation and use them in the quadratic formula.

💡Square Root

The square root operation is a mathematical process that finds a value that, when multiplied by itself, gives the original number. In the quadratic formula, the square root is used to calculate the discriminant (b^2 - 4ac), which determines the nature of the roots of the equation. The script illustrates this with the example where the square root of 25 is 5.

💡Discriminant

The discriminant is the part of the quadratic formula under the square root, represented as (b^2 - 4ac). It is used to determine the number and type of solutions a quadratic equation will have. The video script shows how to calculate the discriminant and its role in finding the roots of the equation.

💡Plus/Minus Symbol

The plus/minus symbol (±) in the quadratic formula indicates that there are two possible solutions to the equation, one by adding the square root of the discriminant to -b, and another by subtracting it. The video script demonstrates this by splitting the formula into two separate solutions for x.

💡Solving for x

Solving for x means finding the values of the variable x that satisfy the quadratic equation. The video script provides a step-by-step guide on how to use the quadratic formula to find these values, emphasizing the importance of considering both the positive and negative outcomes of the square root operation.

💡Checking the Answer

Checking the answer involves substituting the found values of x back into the original equation to verify if they satisfy the equation, ensuring that the solutions are correct. The video script includes a step where the solution x = -2 is substituted back into the equation to confirm its validity.

💡Reduction of Fractions

Reduction of fractions refers to simplifying fractions to their lowest terms by dividing both the numerator and the denominator by their greatest common divisor. In the video, after calculating the solutions, the script shows how to simplify the fractions 2/4 and 8/4 to 1/2 and -2, respectively.

💡Example

An example in the context of the video is a specific quadratic equation used to illustrate the process of solving using the quadratic formula. The script provides two examples, the first being 2x^2 + 3x - 2 = 0 and the second with coefficients a = 6, b = -17, and c = 12, to demonstrate the application of the formula.

Highlights

Introduction to solving quadratic equations using the quadratic formula.

Explanation of the quadratic formula: negative b plus or minus the square root of b squared minus 4ac, divided by 2a.

Identification of coefficients a, b, and c in the standard quadratic equation format.

Demonstration of solving the first example equation: 2x^2 + 3x - 2 = 0.

Calculation of the discriminant (b^2 - 4ac) for the first example.

Finding the square root of the discriminant and simplifying the expression.

Solving for x by considering both the positive and negative parts of the quadratic formula.

Deriving the two solutions for x: 1/2 and -2.

Verification of the solution by plugging -2 back into the original equation.

Introduction of a second example with a different quadratic equation.

Identification of coefficients a, b, and c for the second example: 6, -17, and 12 respectively.

Application of the quadratic formula to the second example.

Calculation of the discriminant for the second example and simplification.

Deriving the two solutions for x: 3/2 and 4/3 from the second example.

Explanation of how to reduce fractions to find the simplified solutions.

Conclusion summarizing the method for solving quadratic equations using the quadratic formula.

Transcripts

in this video we're going to talk about

how to solve quadratic equations

using the quadratic formula

so let's start with this one

let's say we have the equation 2x

squared plus 3x

minus 2 is equal to 0. and our goal is

to solve for x we want to calculate the

value of x that makes this equation true

so here is the quadratic formula that we

need to use

it's negative b

plus or minus the square root

of b squared minus 4ac

divided by 2a

now we need to know what a b and c are

equal to

so in this format where you have all of

the x variables to the left and 0 on the

right

a

is the number in front of x squared

b is the number in front of x and c is

the constant term

so this is going to be

i'm going to rewrite it here

x is equal to negative b b is positive

three

plus or minus the square root of b

squared so that's three squared minus

four

times a a is two times c which is

negative two

all divided by

two a

or two times two

so we have negative three plus or minus

the square root three squared is nine

negative four times two is negative 8

and negative 8 times negative 2 is

positive 16.

on the bottom we have 2 times 2 which is

4.

now 9 plus 16 is 25

and the square root of 25

is 5.

so right now that's what we have

notice the plus or minus symbol so we

need to break this up into two parts

so we're going to have negative 3 plus 5

divided by four

and negative three minus five divided by

four

negative three plus five

is positive two

negative three minus five is negative

eight

so right now we have two

different answers

now we can reduce two over four

to one over two if you divide both

numbers by 2

and 8 divided by 4 is negative 2.

so x

can equal 1 half

or

x can equal

negative 2.

and so that's how you can solve a

quadratic equation using the quadratic

formula

now if you want to check your answer

you can plug it in

let's plug in negative two into this

equation

so we have two times negative two

squared

plus three times minus two

minus two let's see if that equals zero

negative two squared is negative two

times negative two which is four

three times negative two is negative six

now two times four is eight

negative six minus two is negative eight

eight minus eight is zero

so we know that this answer works and

you could try the other one too that's

gonna work as well

but now let's move on to our next

example

let's say we have this particular

quadratic equation

go ahead and use the quadratic formula

to get the answer

so we can see that a is 6

b is negative 17 and c is 12.

so let's begin by writing the formula

so it's x is equal to negative b plus or

minus the square root of b squared minus

4ac

divided by 2a

so b is negative 17.

and then we have b squared that's

negative 17 squared

minus four

a is six

c is 12

divided by two a or two times six

so we have negative times negative 17

that becomes positive 17

negative 17 squared

is going to be positive 289

and then we have negative 4 times 6

which is negative 24

times 12

that's going to be negative 288

2 times 6 is 12.

and inside the square root symbol we

have 289 minus 288

which is the square root of one

and the square root of one is one

so this is what we now have

so we have seventeen

plus one over twelve

at this point when you have the plus and

minus symbol you can break it up into

two answers

and the other answer is going to be

17 minus 1 over 12.

17 plus 1 is 18

and 17 minus 1 is 16.

so now

we just need to reduce

those fractions

so 18 is 6 times 3

twelve is six times two

canceling the six

we get one of our solutions as three

over two

for the other one

sixteen we can write that as four times

four twelve is four times three

so canceling the four

we get the other answer which is four

over three

and so that's it for this video now you

know how to use the quadratic formula to

solve a quadratic equation thanks again

for watching