“The product of two linear expressions is quadratic but the sum of two linear expressions is still linear”.  Do you agree with the statement? Show an example to support your answer.

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degeneratecircle's profile pic

degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted on

The first part is true but the second part isn't. Let `ax+b` and `cx+d` be linear expressions. Their product is `(ac)x^2+(ad+bc)x+db.` Now neither `a` or `c` is zero, or else at least one of `ax+b` and `cx+d` would be constant, not linear. Therefore, `ac!=0,` and the product is quadratic.

The second part isn't true because `x+1` and `-x+1` are both linear expressions, but their sum is `2,` which is constant but not linear.


BUT, I couldn't find a consistent definition for "linear expression" online, so I used the definition in the link below. If something like `2` is considered a linear expression (since the graph of `y=2` is a line, after all), then the sum of two linear expressions would always be linear, but the product of the linear expressions `2` and `3` would almost certainly not be considered "quadratic", so the answer is completely switched. You'll have to check your book or your notes to see exactly what is meant by linear expression. At least in either case, the statement is false.

Sources:
raingal's profile pic

raingal | Student | (Level 1) Honors

Posted on

Yes, I agree with the statement.

For example, here is 2 linear expressions.

x+3=0----(1)

x+7=0----(2)

Sum of the 2 linear expressions:

  x+3=0

+x+7=0

---------

2x+10=0

_________________________

However, product of the 2 linear expressions:

(x+3)(x+7)=x^2+10x+21

____________________________________

This is because the addition of x+x would be 2x, and the equation would stil be a linear equation. However, the product of x*x would be equals x^2, and thus the whole equation becomes a quadratic equation.

Hope this helps:D

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