How To Solve Quadratic Equations Using The Quadratic Formula
Summary
TLDRThis 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
📚 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.
📝 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.
Mindmap
Keywords
💡Quadratic Equations
💡Quadratic Formula
💡Coefficients
💡Square Root
💡Discriminant
💡Plus/Minus Symbol
💡Solving for x
💡Checking the Answer
💡Reduction of Fractions
💡Example
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
تصفح المزيد من مقاطع الفيديو ذات الصلة
5.0 / 5 (0 votes)