Question

If equation 2(x^2 - 4x - 21) - (x-7)(x+77) and (x-7)(x+k) are equivalent, what must be the value of k?

Option a: 83
Option b: 160
Option c: -148
Option d: -71

Answer 1

To find the value of k for equivalent expressions 2(x^2 - 4x - 21) - (x-7)(x+77) and (x-7)(x+k), simplify and compare terms to find that k equals -71.

Step-by-step explanation:

We are given the equation 2(x^2 - 4x - 21) - (x-7)(x+77) and are told that it is equivalent to (x-7)(x+k). The goal is to find the value of k. To determine k, we need to simplify the given equations and compare the coefficients.

Let's expand and simplify the given equations:

1. 2(x^2 - 4x - 21) becomes 2x^2 - 8x - 42.
2. Expanding (x-7)(x+77) gives x^2 + 70x - 539.
3. Subtracting this from the previous result yields 2x^2 - 8x - 42 - (x^2 + 70x - 539), which simplifies to x^2 - 78x + 497.
4. Now, equate that to (x-7)(x+k) which expands to x^2 + (k - 7)x - 7k.

The equation x^2 - 78x + 497 needs to be equivalent to x^2 + (k - 7)x - 7k. Comparing coefficients, we get k - 7 = -78 and -7k = 497. Solving the former for k, we find that k = -71. Therefore, the value of k must be -71 which matches option d.