Final answer:
The value of k for the quadratic expression 2x² + 5x - 12 rewritten in the form (2x-3)(x+k) is 3, which is found through factoring techniques by finding two numbers that multiply to -24 and add to 5, which are 6 and -4.
Step-by-step explanation:
To determine the value of k for the expression 2x² + 5x - 12 rewritten in the form (2x-3)(x+k), we need to use factoring techniques. Factoring a quadratic expression involves breaking it down into two binomials that when multiplied give back the original expression.
Let's start by finding two numbers that multiply to give the product of the coefficient of x² (which is 2) and the constant term (which is -12), and add up to the coefficient of x (which is 5). These two numbers are 6 and -4, since (2)(-12) = -24 and 6 + (-4) = 2.
Now we rewrite the middle term using these two numbers: 2x² + 6x - 4x - 12. Next, we group the terms and factors by grouping:
2x(x+3) - 4(x+3)
Now, we factor out the common binomial (x+3):
(2x - 4)(x + 3)
To have the expression in the desired form (2x-3)(x+k), we can divide the term 2x - 4 by 2, giving us:
(2x - 2*2)(x + 3) = (2x - 3)(x + 3)
Therefore, the value of k is 3.