The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x2).
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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 lượt thích this:
Here are some examples:
| 2x2 + 5x + 3 = 0 | In this one a=2, b=5 và c=3 | |
| x2 − 3x = 0 | This one is a little more tricky: Where is a? Well a=1, as we don"t usually write "1x2" b = −3 và 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) |
Have a Play With It
Play with the "Quadratic Equation Explorer" so you can see:
the function"s graph, và the solutions (called "roots").Hidden Quadratic Equations!
As we saw before, the Standard Form of a Quadratic Equation is
| x2 = 3x − 1 | Move all terms khổng lồ left hand side | x2 − 3x + 1 = 0 | a=1, b=−3, c=1 |
| 2(w2 − 2w) = 5 | Expand (undo the brackets),and move 5 khổng lồ left | 2w2 − 4w − 5 = 0 | a=2, b=−4, c=−5 |
| z(z−1) = 3 | Expand, và move 3 lớn left | z2 − z − 3 = 0 | a=1, b=−1, c=−3 |
The "solutions" lớn the Quadratic Equation are where it is equal to lớn zero.
They are also called "roots", or sometimes "zeros"
There are usually 2 solutions (as shown in this graph).
And there are a few different ways to find the solutions:
Or we can use the special Quadratic Formula:
Just plug in the values of a, b & c, & 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:
x = −b + √(b2 − 4ac) 2a
x = −b − √(b2 − 4ac) 2a
Here is an example with two answers:
But it does not always work out like that!
Imagine if the curve "just touches" the x-axis.Or imagine the curve is so high it doesn"t even cross the x-axis!This is where the "Discriminant" helps us ...
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 it is zero we get just ONE real solution (both answers are the same)
Complex solutions? Let"s talk about them after we see how khổng lồ use the formula.
Using the Quadratic Formula
Just put the values of a, b và 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 & 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
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
A kind reader suggested singing it to lớn "Pop Goes the Weasel":
| ♫ | "x is equal khổng lồ 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 & it will get stuck in your head!
Or you can remember this story:
x = −b ± √(b2 − 4ac) 2a
"A negative boy was thinking yes or no about going khổng lồ a party,at the party he talked to lớn a square boy but not to the 4 awesome chicks.It was all over at 2 am."
Complex Solutions?
When the Discriminant (the value b2 − 4ac) is negative we get a pair of 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
√(−16) = 4i(where i is the imaginary number √−1)
So:x = −2 ± 4i 10
Answer: x = −0.2 ± 0.4i
The graph does not cross the x-axis. That is why we ended up with complex numbers.
In a way it is easier: we don"t need more calculation, we leave it as −0.2 ± 0.4i.
Example: Solve x2 − 4x + 6.25 = 0
Coefficients are:a=1, b=−4, c=6.25
Note that the Discriminant is negative:b2 − 4ac = (−4)2 − 4×1×6.25 = −9
Use the Quadratic Formula:x = −(−4) ± √(−9) 2
√(−9) = 3i(where i is the imaginary number √−1)
So:x = 4 ± 3i 2
Answer: x = 2 ± 1.5i
The graph does not cross the x-axis. That is why we ended up with complex numbers.
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BUT an upside-down mirror image of our equation does cross the x-axis at 2 ± 1.5 (note: missing the i).
Just an interesting fact for you!
Summary
Quadratic Equation in Standard Form: ax2 + bx + c = 0 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 solutions360, 361, 1201, 1202, 2333, 2334, 3894, 3895, 2335, 2336
Quadratic Equation Solver Factoring Quadratics Completing the Square Graphing Quadratic Equations Real World Examples of Quadratic Equations Derivation of Quadratic Equation Algebra Index