Final answer:
The correct factored form of the quadratic equation 2x^2 - 13x - 7 is (2x + 1)(x - 7), which is found by searching for two numbers that multiply to -14 and add to -13, leading to the factors -14 and 1.
Step-by-step explanation:
To factor the quadratic equation 2x^2 - 13x - 7, we look for two numbers that multiply to the product of the coefficient of x^2 (which is 2) and the constant term (which is -7), and add up to the coefficient of x (which is -13).
The two numbers that satisfy these conditions are -14 and 1 because:
- (2)(-7) = -14
- 1(-7) = -7
- -14 + 1 = -13
Now we can rewrite the middle term using these two numbers:
2x^2 - 14x + x - 7
Next, we group the terms and factor by grouping:
(2x^2 - 14x) + (x - 7)
2x(x - 7) + 1(x - 7)
Finally, we factor out the common binomial factor (x - 7):
(2x + 1)(x - 7)
Therefore, the correct factored form of the equation is option c. (2x + 1)(x - 7).