Alright, let's solve the quadratic equation [tex]\( x^2 - 8x + 16 = 0 \)[/tex] step-by-step:
1. Identify the coefficients:
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. By comparing the given equation [tex]\( x^2 - 8x + 16 = 0 \)[/tex] with the standard form, we can identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -8 \)[/tex]
- [tex]\( c = 16 \)[/tex]
2. Solve the quadratic equation:
To find the solutions for the equation, we can use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
However, in this particular case, we notice that the given quadratic equation is a perfect square quadratic, which can be factored as follows:
[tex]\[
x^2 - 8x + 16 = (x - 4)^2 = 0
\][/tex]
3. Find the roots:
Since [tex]\( (x - 4)^2 = 0 \)[/tex], we can solve for [tex]\( x \)[/tex] by setting the inside of the square to zero:
[tex]\[
x - 4 = 0
\][/tex]
Solving this, we get:
[tex]\[
x = 4
\][/tex]
So, the solution to the quadratic equation [tex]\( x^2 - 8x + 16 = 0 \)[/tex] is [tex]\( x = 4 \)[/tex]. This means we have a repeated root (also called a double root) at [tex]\( x = 4 \)[/tex].
To summarize:
[tex]\[
\begin{align*}
a &= 1 \\
b &= -8 \\
c &= 16 \\
\text{Solution} &= [4]
\end{align*}
\][/tex]
