To solve the quadratic equation [tex]\(4x^2 - 32x + 64 = 0\)[/tex], we can use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = 4\)[/tex], [tex]\(b = -32\)[/tex], and [tex]\(c = 64\)[/tex].
1. First, identify the coefficients:
- [tex]\(a = 4\)[/tex]
- [tex]\(b = -32\)[/tex]
- [tex]\(c = 64\)[/tex]
2. Calculate the discriminant [tex]\(\Delta\)[/tex]:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values:
[tex]\[
\Delta = (-32)^2 - 4 \cdot 4 \cdot 64
\][/tex]
[tex]\[
\Delta = 1024 - 1024
\][/tex]
[tex]\[
\Delta = 0
\][/tex]
3. Since the discriminant is zero [tex]\(\Delta = 0\)[/tex], there is exactly one real root for the quadratic equation.
4. Use the quadratic formula to find the root:
[tex]\[
x = \frac{-b \pm \sqrt{0}}{2a}
\][/tex]
[tex]\[
x = \frac{-(-32)}{2 \cdot 4}
\][/tex]
[tex]\[
x = \frac{32}{8}
\][/tex]
[tex]\[
x = 4
\][/tex]
Therefore, the solution to the equation [tex]\(4x^2 - 32x + 64 = 0\)[/tex] is:
[tex]\[
x = 4.0
\][/tex]
Since the discriminant is 0, the equation has only one unique real solution which is [tex]\(x = 4.0\)[/tex].
