Question

The expressions 2(x² – 4x − 21) – (x − 7) (x +77) and (x − 7) (x + k) are equivalent. – What must be the value of k?​

-71
Step-by-step explanation:
The constant term of is .
The constant term of is .
So, .

the value of k must be 2.
Step-by-step 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.