To solve the quadratic equation x² – 8x – 5 = 0 using the quadratic formula, we start by identifying the coefficients a, b, and c in the standard form ax² + bx + c = 0. Here, we have:
- a = 1
- b = -8
- c = -5
The quadratic formula is:
x = (-b ± √(b² – 4ac)) / (2a)
Now, we need to calculate the discriminant (the part under the square root):
b² – 4ac = (-8)² – 4(1)(-5) = 64 + 20 = 84
Since the discriminant is positive, we will have two real and distinct solutions. Now we can plug our values into the quadratic formula:
x = (8 ± √84) / (2 * 1)
We simplify this further:
√84 simplifies to 2√21, so we get:
x = (8 ± 2√21) / 2
Now, we can divide each term by 2:
x = 4 ± √21
Thus, the two solutions are:
x₁ = 4 + √21
x₂ = 4 – √21
These represent the two points where the graph of the equation intersects the x-axis.