Quadratic formula

     

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 like this:

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a
, bc are known values. a can"t be 0.

Here are some examples:


2x2 + 5x + 3 = 0 In this one a=2, b=5 & 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 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)

Have a Play With It

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

the function"s graph, and the solutions (called "roots").

Hidden Quadratic Equations!

As we saw before, the Standard Form of a Quadratic Equation is


In disguise
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In Standard size a, b and c
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 lớn left 2w2 − 4w − 5 = 0 a=2, b=−4, c=−5
z(z−1) = 3 Expand, & move 3 to left z2 − z − 3 = 0 a=1, b=−1, c=−3

The "solutions" to lớn the Quadratic Equation are where it is equal 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 lớn find the solutions:


Or we can use the special Quadratic Formula:

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Just plug in the values of a, b and 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 lượt thích ± ?

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 lượt thích 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 to use the formula.

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Using the Quadratic Formula

Just put the values of a, b và c into the Quadratic Formula, & 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

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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 khổng lồ "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 and 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 to a party,at the các buổi party he talked to a square boy but not lớn 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

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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

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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 solutions


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