-x²-4x+5>0
To solve the inequality -x² - 4x + 5 > 0, we first need to find the critical points of the quadratic function by setting the expression equal to zero:
-x² - 4x + 5 = 0
Next, we can solve for x by factoring or using the quadratic formula:
-x² - 4x + 5 = 0 -(x² + 4x - 5) = 0 -(x - 1)(x + 5) = 0
Setting each factor to zero gives us: x - 1 = 0 or x + 5 = 0 x = 1 x = -5
Therefore, the critical points are x = 1 and x = -5.
Now we need to determine the sign of the quadratic expression in the intervals created by the critical points:
For x < -5: Substitute x = -6 into the expression: -(-6)² - 4(-6) + 5 = -36 + 24 + 5 = -7 Since -7 is less than 0, the expression is negative in this interval.
For -5 < x < 1: Substitute x = 0 into the expression: -(0)² - 4(0) + 5 = 5 Since 5 is greater than 0, the expression is positive in this interval.
For x > 1: Substitute x = 2 into the expression: -(2)² - 4(2) + 5 = -4 - 8 + 5 = -7 Since -7 is less than 0, the expression is negative in this interval.
Therefore, the solution to the inequality -x² - 4x + 5 > 0 is x < -5 or x > 1.
