Question

PLEASE HELP
Consider the quadratic expression $13x^2 + nx - 17.$ For certain values of $n,$ it may be factored into a product of two linear polynomials, both of which have integer coefficients. What are all such values of $n?$

Answer 1

A quadratic equation that is factored into linear polynomials is of the form $(ax + b) \times (cx + d)$ where a, b, c and d are integers. The possible values of n are 220, -220, -4 and 4.

Let the expression be:

So:

$13x^2 + nx - 17 = (ax + b) \times (cx + d)$

Multiply the coefficients of x^2 (13) and the constant (-17)

$k = 13 \times -17$

$k = -221$

List out all prime factors of k

$Factors = \{(-1,221),(1,-221),(13,-17),(-13,17)\}$

Add the factors to give the possible values of n

$n =\{-1+221,1-221,13-17,-13+17\}$

$n = \{220,-220,-4,4\}$

Hence, the possible values of n are 220, -220, -4 and 4

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