Let's solve the quadratic equation [tex]\( x^2 - 8x + 16 = 0 \)[/tex].
1. Identify coefficients:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -8\)[/tex]
- [tex]\(c = 16\)[/tex]
2. Formulate the quadratic formula:
- The quadratic formula is [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex].
3. Calculate the discriminant:
- The discriminant [tex]\(\Delta\)[/tex] of the quadratic equation [tex]\( ax^2 + bx + c \)[/tex] is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
- For [tex]\( x^2 - 8x + 16 = 0 \)[/tex]:
[tex]\[
\Delta = (-8)^2 - 4 \cdot 1 \cdot 16
\][/tex]
[tex]\[
\Delta = 64 - 64
\][/tex]
[tex]\[
\Delta = 0
\][/tex]
4. Evaluate the discriminant:
- If [tex]\(\Delta > 0\)[/tex], there are two distinct real solutions.
- If [tex]\(\Delta = 0\)[/tex], there is exactly one real solution.
- If [tex]\(\Delta < 0\)[/tex], there are no real solutions.
5. Solve for [tex]\(x\)[/tex]:
- Since [tex]\(\Delta = 0\)[/tex], there is exactly one real solution.
- Using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
- Substitute the known values:
[tex]\[
x = \frac{-(-8) \pm \sqrt{0}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{8 \pm 0}{2}
\][/tex]
[tex]\[
x = \frac{8}{2}
\][/tex]
[tex]\[
x = 4
\][/tex]
Therefore, the solution to the quadratic equation [tex]\( x^2 - 8x + 16 = 0 \)[/tex] is:
[tex]\[
x = 4
\][/tex]
