Final answer:
To find the values of x for which the given expression is negative, we solve the inequality by factoring the expression and considering the sign of each factor. The expression x^4 - 51x^2 + 50 is negative for a total of 12 integers.
Step-by-step explanation:
To find the values of x for which the given expression is negative, we can solve the inequality x^4 - 51x^2 + 50 < 0. Let's factor this expression: (x^2 - 1)(x^2 - 50) < 0. Now we have two factors, so we need to consider the sign of each factor separately.
For the first factor, (x^2 - 1), we can set it equal to zero and solve for x: x^2 - 1 = 0. Solving this quadratic equation gives us x = -1 and x = 1. Since it's a less than inequality, we need the intervals where the factor is negative. So the first factor is negative for x < -1 and -1 < x < 1.
For the second factor, (x^2 - 50), we can set it equal to zero and solve for x: x^2 - 50 = 0. Solving this quadratic equation gives us x = -√50 and x = √50. Since it's a less than inequality, we need the intervals where the factor is negative. So the second factor is negative for x < -√50 and √50 < x.
Now we need to find the values of x that satisfy both conditions. We need to find the intersection of the intervals for each factor. From the first factor, we have (-∞, -1) U (1, ∞). From the second factor, we have (-√50, √50). Combining these intervals, we get (-∞, -√50) U (-1, √50) U (1, ∞). These are the intervals where the expression x^4 - 51x^2 + 50 is negative. To find the number of integers in these intervals, we can count the numbers between -√50 and -1, between -1 and √50, and between √50 and ∞. Counting these intervals, we get a total of 12 integers.