Final answer:
To find an integer n such that the interval of the form [n, n+1] contains a solution to the equation x² - x = 1, we need to find a value of n for which the equation is satisfied.
Step-by-step explanation:
To find an integer n such that the interval of the form [n, n+1] contains a solution to the equation x² - x = 1, we need to find a value of n for which the equation is satisfied. Let's solve the equation:
x² - x = 1
x² - x - 1 = 0
Now, we can use the quadratic formula to find the solutions:
x = (-b ± √(b² - 4ac)) / (2a)
For this equation, a = 1, b = -1, and c = -1.
Substituting these values into the formula, we get:
x = (-(-1) ± √((-1)² - 4(1)(-1))) / (2(1))
x = (1 ± √(1 + 4)) / 2
x = (1 ± √5) / 2
Therefore, for the interval [n, n+1] to contain a solution to the equation, n can be any integer such that n ≤ (1 + √5) / 2.