Solving the Equation x² – 20x + 100 = 36
To solve the equation x² – 20x + 100 = 36, we first need to rearrange it in the standard quadratic form.
1. **Rearranging the equation:**
We subtract 36 from both sides, giving us:
x² – 20x + 100 – 36 = 0.
This simplifies to:
x² – 20x + 64 = 0.
2. **Using the quadratic formula:**
The quadratic formula is given by x = (-b ± √(b² – 4ac)) / (2a), where a, b, and c are the coefficients from the equation ax² + bx + c = 0.
For our equation, we have:
– a = 1
– b = -20
– c = 64
Plugging these values into the formula yields:
x = (20 ± √((-20)² – 4 * 1 * 64)) / (2 * 1)
3. **Calculating the discriminant:**
Calculate (-20)² – 4 * 1 * 64 = 400 – 256 = 144. Since the discriminant (the term inside the square root) is positive, we will have two real solutions.
4. **Finding the solutions:**
Now substituting back, we get:
x = (20 ± √144) / 2
Since √144 = 12, the equation becomes:
x = (20 ± 12) / 2.
Calculating the two potential solutions:
– First solution:
x = (20 + 12) / 2 = 32 / 2 = 16
– Second solution:
x = (20 – 12) / 2 = 8 / 2 = 4
5. **Summary of the solutions:**
The values of x that solve the equation x² – 20x + 100 = 36 are x = 16 and x = 4.
Additionally, you mentioned alternate factors such as (x – 4)(x – 16) = 0, (x – 8)(x – 4) = 0, and (x – 16) = 0. This clearly aligns with our two solutions of x = 4 and x = 16.