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.

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Uniteasy Admin · Accepted Answer

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)
$$96 =2	imes48$$
$$=2	imes2	imes24$$
$$=2	imes2	imes2	imes12$$
$$=2	imes2	imes2	imes2	imes6$$
$$=2	imes2	imes2	imes2	imes2	imes 3$$
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 $$k$$
$$k = \{\pm2,\pm5,\pm10,\pm23 \}$$

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