Select the values of x that are solutions to the inequality 0 > x^2 + 5x -2

To find the values of x that are solutions to the inequality 0 > x^2 + 5x - 2, we need to solve the quadratic equation x^2 + 5x - 2 = 0 first.

To solve the quadratic equation, we can use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac))/(2a)

In our case, a = 1, b = 5, and c = -2. Plugging these values into the quadratic formula, we get:

x = (-(5) ± √((5)^2 - 4(1)(-2)))/(2(1))

Simplifying further, we have:

x = (-5 ± √(25 + 8))/2

x = (-5 ± √33)/2

So the two potential solutions for x are:

x = (-5 + √33)/2 and x = (-5 - √33)/2

Now we need to determine which values of x satisfy the original inequality 0 > x^2 + 5x - 2.

We know that in order for the inequality to be true, x^2 + 5x - 2 must be negative.

Let's consider the first potential solution x = (-5 + √33)/2:

Substituting this value into the equation x^2 + 5x - 2, we have:

((-5 + √33)/2)^2 + 5((-5 + √33)/2) - 2

Simplifying further, we get:

(25 - 10√33 + 33)/4 - (25 - 5√33)/2 - 2

(25 - 10√33 + 33 - (50 - 10√33) - 8)/4

(58 - 50)/4

8/4

2

Since 2 is greater than 0 and not less than 0, the value of x = (-5 + √33)/2 is not a solution to the inequality.

Now let's consider the second potential solution x = (-5 - √33)/2:

Substituting this value into the equation x^2 + 5x - 2, we have:

((-5 - √33)/2)^2 + 5((-5 - √33)/2) - 2

Simplifying further, we get:

(25 + 10√33 + 33)/4 - (25 + 5√33)/2 - 2

(25 + 10√33 + 33 - (50 + 10√33) - 8)/4

(58 - 50)/4

8/4

2

Similarly, since 2 is greater than 0 and not less than 0, the value of x = (-5 - √33)/2 is also not a solution to the inequality.

Therefore, there are no solutions to the inequality 0 > x^2 + 5x - 2.