To find the values of x for the equation x² – 2x – 20 = 0, we can use the quadratic formula:
x = (-b ± √(b² – 4ac)) / (2a)
In this equation, a, b, and c are the coefficients from our standard quadratic equation ax² + bx + c = 0. For our equation:
- a = 1
- b = -2
- c = -20
Now, we need to calculate the discriminant, which is b² – 4ac:
Discriminant = (-2)² – 4(1)(-20)
Discriminant = 4 + 80 = 84
Since the discriminant is positive (84), we have two real distinct solutions. Now, we apply the values into the quadratic formula:
x = (2 ± √84) / 2
Next, we simplify √84:
√84 = √(4 × 21) = 2√21
Substituting back into our equation gives:
x = (2 ± 2√21) / 2
Now we can simplify further:
x = 1 ± √21
Thus, the two values for x are:
- x = 1 + √21
- x = 1 – √21
These solutions represent the x-values where the parabola represented by the equation intersects the x-axis.