#### Question

Find all the values of k so that the quadratic expression 3x^2 + kx - 8 factors into two binomials. Explain the process used to find the values.

Question

Find all the values of k so that the quadratic expression 3x^2 + kx - 8 factors into two binomials. Explain the process used to find the values.

For the quadratic expression 3x^2 + kx - 8, the zeros could be determined by the root formula,

\dfrac{-k\pm\sqrt{k^2+96}}{6}

In order for the expression to be able to factor to two binomials, the discriminant must be a perfect square.

Let a = \sqrt{k^2+96}

Square both sides

a^2 = k^2+96

Isolate the constant, then

a^2-k^2 = 96

Apply difference of squares identity

(a+k)(a-k)=96

The expression shows a+k and a-k is a pair of factors of 96

Let f_1 = a+k, f_2 = a-k

Then k = \dfrac{1}{2}(f_1-f_2)

Now we are going to find the pairs of factors of which the difference is the multiples of 2 (so that a is an integer)

96 =2\times48

=2\times2\times24

=2\times2\times2\times12

=2\times2\times2\times2\times6

=2\times2\times2\times2\times2\times 3

Case 1 k=23 if f_1 = 48 and f_2 = 2

Case 2 k = 10 if f_1 = 24 and f_2 = 4

Case 3 k=2 if f_1 =12 and f_2 = 8

Case 4 k = 5 if f_1 =16 and f_2 = 6

Then k = \{2,5,10,23 \} are set to make the quadratic expression 3x^2 + kx - 8 factor into two binomials. Since the opposite of k will not change the value of the discriminant, the opposite of k are also among the solution set. Therefore, we have the following solution set for k

k = \{\pm2,\pm5,\pm10,\pm23 \}