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 expression written in the form (x +
3)²-6, say, is said to be in completed square form. |
A quadratic function can be written
in this form by completing the square. The following quadratic functions
have all been written in completed square form:
x² + 6x + 3=(x + 3)²-6
x² - 8x + 2=(x - 4)²-14
x² - 10x - 1=(x - 5)²-26
Example: write in completed square form: 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 |
|
