Question

If a(x) and b(x) are linear functions with one variable, which of the following expressions produces a quadratic function?
(ab)(x)
mc012-1.jpg
(a – b)(x)
(a + b)(x)

Answer 1

Linear × Linear = Quadratic
So the correct answer for this question is
(ab)(x)

Answer 2

First, let us clearly differentiate between a linear equation and a quadratic equation.
An equation having degree (highest power) 1 is called a linear equation, whereas an equation having degree 2 is a quadratic equation.

Coming back to the question, let us consider each type of operation and then conclude with our answer.

First, let us consider addition. Addition of two linear equations results in a linear equation itself. 
For example, let a(x) be x+2 and b(x) be x+5. Then, (a+b)(x) is (x+2)+(x+5) which is equal to 2x+7, which is also a linear equation.

Now, coming to subtraction, even this results in a linear equation.
Taking a(x) to be 2x+5 and b(x) to be x+3, (a-b)(x) will be (2x+5)-(x+3) which is equal to x+2, which is still a linear equation.

The only exception, however, is multiplication. Multiplication of two linear equations yields a quadratic equation.
Taking a(x) = x+1 and b(x) = x+2, we see that (ab)(x) is equal to (x+1)(x+2) which is equal to $x^{2}$ + 3x + 2. As there is an $x^{2}$ term, it is a quadratic equation.

Division is quite opposite, as it reduces the degree. 
For example, if a(x) is x+1 and b(x) is also x+1, then (a/b)(x) is equal to 1, which is devoid of terms containing x.

Thus, only (ab)(x) produces a quadratic equation.