D2. Elementary algebra : completing the square.
Vocabulary
Algebraic expressions like x2+3x+6 or x2-5x+11 are called
quadratic. Any expression that can be written in the form ax2+bx+c,
where a, b and c are numbers, is called a quadratic expression. |
Any quadratic function can be written in a form known as
completing the square. The following quadratic functions have all been
written in this form:
x2 + 6x + 3=(x + 3)2-6
x2 - 8x + 2=(x - 4)2-14
x2 - 10x - 1=(x - 5)2-26
Example: complete the square with x2 + 4x +
3
| Step 1: |
Write the expression in the required form |
x2 + 4x + 3=(x + ?)2+?
|
| Step 2: |
The first unknown ? must be half the coefficient of
x: |
x2 + 4x + 3=(x + 2)2+?
|
| Step 3 |
The second unknown ? can be found by multiplying
out |
(x + 2)2+?=x2 +4x + 4 + ?
|
| Step 4 |
Compare constants |
3=4 + ? |
| Step 5 |
The second ? can be calculated: |
?=-1 |
| |
So the answer is found. |
x2 + 4x + 3=(x + 2)2-1
|
|
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