The quadratic formula is a useful tool for finding the solutions to any quadratic equation of the form ax² + bx + c = 0. The formula is:
x = (-b ± √(b² – 4ac)) / (2a)
In this case, your equation is:
x² – 2x – 20 = 0
Here, we can identify the coefficients:
- a = 1
- b = -2
- c = -20
Now, plug these values into the quadratic formula:
x = (-(-2) ± √((-2)² – 4(1)(-20))) / (2(1))
First, simplify what’s under the square root:
b² – 4ac = (-2)² – 4(1)(-20) = 4 + 80 = 84
Now, substitute it back into the formula:
x = (2 ± √84) / 2
Next, simplify √84. Note that √84 = √(4 * 21) = 2√21. So we can rewrite our equation:
x = (2 ± 2√21) / 2
Now, simplify further by dividing all terms by 2:
x = 1 ± √21
Thus, we find two values for x:
- x = 1 + √21
- x = 1 – √21
To get numerical approximations:
- x ≈ 5.58
- x ≈ -3.58
So, the values of x which solve the equation x² – 2x – 20 = 0 are approximately 5.58 and -3.58.