Question

can someone explain quadratic equations?

Answer 1

sure!

Answer 2

google

Answer 3

Sure! What about them do you need explained?

Answer 4

An example of a Quadratic Equation:

Quadratic Equation

Quadratic Equations make nice curves, like this one:



Name

The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x2).
It is also called an "Equation of Degree 2" (because of the "2" on the x)
Standard Form

The Standard Form of a Quadratic Equation looks like this:

Quadratic Equation
a, b and c are known values. a can't be 0.
"x" is the variable or unknown (we don't know it yet).
Here are some more examples:

2x2 + 5x + 3 = 0 In this one a=2, b=5 and c=3

x2 − 3x = 0 This one is a little more tricky:
Where is a? Well a=1, and we don't usually write "1x2"
b = -3
And where is c? Well c=0, so is not shown.
5x − 3 = 0 Oops! This one is not a quadratic equation: it is missing x2
(in other words a=0, which means it can't be quadratic)
Hidden Quadratic Equations!

So the "Standard Form" of a Quadratic Equation is

ax2 + bx + c = 0

But sometimes a quadratic equation doesn't look like that! For example:

In disguise → In Standard Form a, b and c
x2 = 3x -1 Move all terms to left hand side x2 - 3x + 1 = 0 a=1, b=-3, c=1
2(w2 - 2w) = 5 Expand (undo the brackets),
and move 5 to left 2w2 - 4w - 5 = 0 a=2, b=-4, c=-5
z(z-1) = 3 Expand, and move 3 to left z2 - z - 3 = 0 a=1, b=-1, c=-3
5 + 1/x - 1/x2 = 0 Multiply by x2
5x2 + x - 1 = 0 a=5, b=1, c=-1


Quadratic Graph
Have a Play With It

Play with the "Quadratic Equation Explorer" so you can see:

the graph it makes, and
the solutions (called "roots").
How To Solve It?

The "solutions" to the Quadratic Equation are where it is equal to zero.
There are usually 2 solutions (as shown in the graph above).
They are also called "roots", or sometimes "zeros"
There are 3 ways to find the solutions:

1. We can Factor the Quadratic (find what to multiply to make the Quadratic Equation)
2. We can Complete the Square, or
3. We can use the special Quadratic Formula:
Quadratic Formula

Just plug in the values of a, b and c, and do the calculations.

We will look at this method in more detail now.

About the Quadratic Formula

Plus/Minus

First of all what is that plus/minus thing that looks like ± ?



The ± means there are TWO answers:



Here is why we can get two answers:

Quadratic Graph
But sometimes we don't get two real answers, and the "Discriminant" shows why ...

Discriminant

Do you see b2 - 4ac in the formula above? It is called the Discriminant, because it can "discriminate" between the possible types of answer:

when b2 - 4ac is positive, we get two Real solutions
when it is zero we get just ONE real solution (both answers are the same)
when it is negative we get two Complex solutions
Complex solutions? Let's talk about them after we see how to use the formula.



Using the Quadratic Formula

Just put the values of a, b and c into the Quadratic Formula, and do the calculations.

Example: Solve 5x² + 6x + 1 = 0

Coefficients are: a = 5, b = 6, c = 1

Quadratic Formula: x = [ -b ± √(b2-4ac) ] / 2a

Put in a, b and c: x = [ -6 ± √(62-4×5×1) ] / (2×5)

Solve: x = [ -6 ± √(36-20) ]/10
x = [ -6 ± √(16) ]/10
x = ( -6 ± 4 )/10
x = -0.2 or -1


5x^2+6x+1
Answer: x = -0.2 or x = -1



And we see them on this graph.



Check -0.2: 5×(-0.2)² + 6×(-0.2) + 1
= 5×(0.04) + 6×(-0.2) + 1
= 0.2 -1.2 + 1
= 0
Check -1: 5×(-1)² + 6×(-1) + 1
= 5×(1) + 6×(-1) + 1
= 5 - 6 + 1
= 0


Remembering The Formula

I don't know of an easy way to remember the Quadratic Formula, but a kind reader suggested singing it to "Pop Goes the Weasel":

♫ "x equals minus b ♫ "All around the mulberry bush
plus or minus the square root The monkey chased the weasel
of b-squared minus four a c The monkey thought 'twas all in fun
all over two a" Pop! goes the weasel"
Try singing it a few times and it will get stuck in your head!

Complex Solutions?

When the Discriminant (the value b2 - 4ac) is negative we get Complex solutions ... what does that mean?

It means our answer will include Imaginary Numbers. Wow!

Example: Solve 5x² + 2x + 1 = 0

Coefficients are: a = 5, b = 2, c = 1

Note that The Discriminant is negative: b2 - 4ac = 22 - 4×5×1 = -16

Use the Quadratic Formula: x = [ -2 ± √(-16) ] / 10

The square root of -16 is 4i
(i is √-1, read Imaginary Numbers to find out more)

So: x = ( -2 ± 4i )/10
5x^2+6x+1
Answer: x = -0.2 ± 0.4i



The graph does not cross the x-axis. That is why we ended up with complex numbers.

In some ways it is easier: we don't need more calculation, just leave it as -0.2 ± 0.4i.



Summary

Quadratic Equation in Standard Form: ax2 + bx + c = 0
Quadratic Equations can be factored
Quadratic Formula: x = [ -b ± √(b2-4ac) ] / 2a
When the Discriminant (b2-4ac) is:
positive, there are 2 real solutions
zero, there is one real solution
negative, there are 2 complex solutions
Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10
(Hard Questions: 1 2 3 4 5 6 7 8 )

Answer 5

In elementary algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the form

ax^2+bx+c=0
where x represents an unknown, and a, b, and c represent known numbers such that a is not equal to 0. If a = 0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term.[1]

Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation only contains powers of x that are non-negative integers, and therefore it is a polynomial equation, and in particular it is a second degree polynomial equation since the greatest power is two.

Answer 6

lol you asked didn't you
google it
go to class if you take one

Answer 7

You should already know how to factor quadratics. (If not, review Factoring Quadratics.) The new thing here is that the quadratic is part of an equation, and you're told to solve for the values of x that make the equation true. Here's how it works:

Solve (x – 3)(x – 4) = 0.
Okay, this one is already factored for me. But how do I solve this?

Think: If I multiply two things together and the result is zero, what can I say about those two things? I can say that at least one of them must also be zero. That is, the only way to multiply and get zero is to multiply by zero. (This is sometimes called "The Zero Factor Property" or "Rule" or "Principle".)

Answer 8

@redheadangel
Why don't you post the link instead of posting huge peices f information that practically spam the post cx

Answer 9

lol sorry :)

Answer 10

It's kk c:

Answer 11

http://www.purplemath.com/modules/solvquad.htm

Answer 12

www.mathsisfun.com/algebra/quadratic-equation.html

Answer 13

en.wikipedia.org/wiki/Quadratic_equation

Answer 14

thank you so much!!!

Answer 15

np!!

Answer 16

this is a good place to start
https://www.khanacademy.org/math/algebra/quadratics/quadratic_odds_ends/v/introduction-to-the-quadratic-equation

Answer 17

scarlet im sorry if it was for something i said.

Answer 18

what?

Answer 19

we were just talking you blocked me.

Answer 20

oops sorry i didnt mean to

Answer 21

its ok stuff happens

Answer 22

friends again

Answer 23

yep :)

Answer 24

can we go back to talkin in the messages this is for the answers.

Answer 25

yes