Section 5.1 Quadratics
In Section 3.3 we learned to identify data that has a quadratic relation. Here we will learn to recognize these in algebraic notation and review their properties.
Subsection 5.1.1 Algebraic Forms of Quadratics
Quadratic refers to any expression or equation that has a non-zero squared term. Table 5.1.1 shows some of the common forms.
| Quadratic | Non-quadratic |
| \(11x^2+32x-3\) | \(5x+3\) |
| \(2(x-3)^2+7\) | \(y=x^3+7x^2-5x+3\) |
| \(y=23-3x^2\) | \(y=\frac{17}{x^2}\) |
| \(x(6x-5)=21\) | \(x^2(x-5)=7\) |
| \(y=(x+3)(x-5)\) | \(y=0x^2+3x+2\) |
The first two forms of quadratics above are common because they are useful for various applications. The first one, called standard form, is written as \(ax^2+bx+c\) where \(a \ne 0\text{.}\) For \(11x^2+32x-3\text{,}\) \(a=11\text{,}\) \(b=32\text{,}\) and \(c=-3\text{.}\) We will use this form in this section for the purpose of solving.
Subsection 5.1.2 Quadratic Pattern
We can show that expressions in these forms produce the data pattern we identified in Example 3.2.5. Table 5.1.2 demonstrates that \(2x^2+3x+2\) has the same pattern with second differences.
| \(x\) | \(2x^2+3x+2\) | 1st Difference | 2nd Difference |
| -3 | 11 | ||
| -2 | 4 | -7 | |
| -1 | 1 | -3 | 4 |
| 0 | 2 | 1 | 4 |
| 1 | 7 | 5 | 4 |
| 2 | 16 | 9 | 4 |
