Answer:
the value of k must be 2.
Explanation:
To find the value of k, we can set the two expressions equal to each other and solve for k. We have:
2(x² - 4x − 21) - (x − 7) (x +77) = (x − 7) (x + k)
Expanding the left side gives:
2x² - 8x - 42 - x² + 7x + 77 = x² - 7x + kx - 7k
Combining like terms on both sides gives:
x² - 15x + 119 - 7k = 0
This is a quadratic equation in the form ax² + bx + c = 0. To solve for x, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Substituting the values for a, b, and c, we get:
x = (15 ± √(225 - 4(1)(119 - 7k))) / 2
Since we are only interested in the value of k, we can disregard the solutions for x. Solving for k, we find that:
k = (-119 + 225 - 4(1)(15)) / (2(-7))
Simplifying this expression gives:
k = (-119 + 225 + 60) / (-14)
k = (-34) / (-14)
k = 34/14
k = 2
Therefore, the value of k must be 2.