To solve the quadratic equation x² – 6x – 22 = 0, we can use the quadratic formula, which is given by:
x = (-b ± √(b² – 4ac)) / (2a)
In this equation:
- a = 1
- b = -6
- c = -22
First, we need to calculate the value of the discriminant, b² – 4ac:
b² = (-6)² = 36
4ac = 4 * 1 * (-22) = -88
Now, substituting these values into the discriminant formula:
Discriminant = 36 – (-88) = 36 + 88 = 124
Since the discriminant is positive (124), we will have two distinct real solutions. Now we can substitute back into the quadratic formula:
x = (6 ± √124) / 2
Simplifying √124, we get:
√124 = √(4 * 31) = 2√31
Now substituting this back, we have:
x = (6 ± 2√31) / 2
Breaking this down, we can simplify:
x = 3 ± √31
Thus, the two solutions for the quadratic equation x² – 6x – 22 = 0 are:
x = 3 + √31 and x = 3 – √31.