To solve the equation x² – 18x + 81 = 25, we first aim to bring all terms of the equation to one side. This forms a standard quadratic equation format. Start by subtracting 25 from both sides:
x² – 18x + 81 – 25 = 0
This simplifies our equation:
x² – 18x + 56 = 0
Next, we will solve the quadratic equation using the quadratic formula, which is x = (-b ± √(b² – 4ac)) / 2a. For our equation, we assign:
- a = 1
- b = -18
- c = 56
Now, let’s calculate the discriminant (b² – 4ac):
(-18)² – 4(1)(56) = 324 – 224 = 100
Since the discriminant is positive, we can find two distinct real solutions. Now, we plug the values into the quadratic formula:
x = (18 ± √100) / 2(1)
Calculating further:
x = (18 ± 10) / 2
This gives us two potential solutions:
- x₁ = (18 + 10) / 2 = 28 / 2 = 14
- x₂ = (18 – 10) / 2 = 8 / 2 = 4
So, the solutions to the equation x² – 18x + 81 = 25 are:
x = 14 and x = 4.