Linear to Quadratic
We know that \(3 \cdot 4 = 12\). We also know that \(2 \cdot 6 = 12\). Also, \(-3 \cdot-4 = 12\) and \(-2 \cdot-6 = 12\). In the case of \(3 \cdot 4 = 12\), 3 and 4 are called factors, while 12 is the product.
Quadratics can be represented as the product of 2 factors also. For example, \(x^2+7x+12\) is the product of \((x+3)(x+4)\). Each of the factors happens to be a linear expression while the product is a quadratic expression.
When we combine expressions or equations, we are building new functions. Sometimes they will stay the same type, such as linear, other times they will change, possibly to quadratic. We can add, subtract, multiply or divide expressions, just like we can with numbers. Take a look at the following to we examples.
Let’s let \(a(x) = (x+3)\) and \(b(x)= (x+4)\) then we can say that
\[\begin{align} a(x) \cdot b(x) &=(x+3)(x+4) \\\\ &=x^2+7x+12 \\\\ &\text{and} \\\\ a(x)+b(x)&=(x+3)+(x+4) \\\\ &=(2x+7)\end{align}\]
What do you notice about \(a(x) \cdot b(x)\) and \(a(x) + b(x)\)?
What do you wonder? Would you agree that \(x\cdot(x+2)\) results in a quadratic? Why?
Put into practice the new concepts we just discussed.
