Final answer:
To determine the values of the variable for which the expression is defined as a real number, use the quadratic formula to solve the quadratic equation and find the values of x. The values of x are 8 and approximately 5.36, giving the solution in interval notation as [5.357142857, 8].
Step-by-step explanation:
To determine the values of the variable for which the expression is defined as a real number, we need to find the values of x that make the quadratic equation 7x² − 65x + 18 equal to a real number. A quadratic equation is defined as a real number when the discriminant (b² - 4ac) is greater than or equal to zero. In this case, the quadratic equation is in the form ax² + bx + c = 0, where a = 7, b = -65, and c = 18. We can use the quadratic formula, x = (-b ± √(b² - 4ac))/(2a), to find the values of x. Plugging in the values, we get:
x = (-(-65) ± √((-65)² - 4(7)(18)))/(2(7))
x = (65 ± √(4225 - 504))/(14)
x = (65 ± √3721)/(14)
x = (65 ± 61)/(14)
x = 8 or x = 5.357142857
In interval notation, the values of x for which the expression is defined as a real number are [5.357142857, 8].