Quadratic Equations
This is what a "Standard" Quadratic Equation looks like:
- The letters a, b and c are coefficients (you know those values). They can have any value, except that a can't be 0.
- The letter "x" is the variable or unknown (you don't know it yet)
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Here is an example of one:
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)
More Examples of Quadratic Equations:
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In this one a=2, b=5 and c=3 |
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This one is a little more tricky:
- Where is a? In fact a=1, as we don't usually write "1x2"
- b = -3
- And where is c? Well, c=0, so is not shown.
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Oops! This one is not a quadratic equation, because it is missing x2 (in other words a=0, and that means it can't be quadratic) |
Hidden Quadratic Equations!
So far we have seen the "Standard Form" of a Quadratic Equation:
ax² + bx + c = 0
But sometimes a quadratic equation doesn't look like that!
Here are some examples of different forms for you:
| 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
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5x2 + x - 1 = 0 |
a=5, b=1, c=-1 |
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:
3. You can use the special 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 ± ?
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The ± means there are TWO answers:
Why two answers? Just look at this typical graph: |
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Discriminant
The blue part (b2 - 4ac) is called the discriminant, because it can "discriminate" between the possible
types of answer:
- when b2 - 4ac is positive, you will get two real solutions
- when it is zero you get just ONE real solution (both answers are the same)
- when it is negative you
get two Complex solutions
I will explain about Complex solutions later.
Remembering
I don't know of an easy way to remember the Quadratic Formula, I just say to myself:
"minus b plus or minus the square root of b-squared minus four ac, all over two a"
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: |
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a = 5, b = 6, c = 1 |
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| Quadratic Formula: |
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x = [ -b ± √(b2-4ac) ] / 2a |
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| Put in a, b and c: |
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x = [ -6 ± √(62-4×5×1) ] / (2×5) |
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| Solve: |
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x = [ -6 ± √(36-20) ]/10 |
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x = [ -6 ± √(16) ]/10 |
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x = ( -6 ± 4 )/10 |
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x = -0.2 or -1 |
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Answer: x = -0.2 or x = -1
And you can 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 |
Complex Solutions?
When the Discriminant (the value b2 - 4ac) is negative you get Complex solutions ... what does that mean?
It means your answer will include Imaginary Numbers. Wow!
Example: Solve 5x² + 2x + 1 = 0
| Coefficients are: |
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a = 5, b = 2, c = 1 |
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| The Discriminant is negative: |
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b2 - 4ac = 22 - 4×5×1 = -16 |
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| Use the Quadratic Formula: |
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x = [ -2 ± √(-16) ] / 10 |
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The square root of -16 is 4i, where i is √-1
(Read Imaginary Numbers to find out why) |
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| So: |
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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 some ways it is actually easier ... you don't have to calculate the two solutions, 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
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