To solve the quadratic equation [tex]\( x^2 = 16x - 65 \)[/tex], we need to rearrange it into its standard form, which is [tex]\( ax^2 + bx + c = 0 \)[/tex].
1. Start with the given equation:
[tex]\[
x^2 = 16x - 65
\][/tex]
2. Rearrange it to standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[
x^2 - 16x + 65 = 0
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -16 \)[/tex], and [tex]\( c = 65 \)[/tex].
3. Use the quadratic formula to find the solutions for [tex]\( x \)[/tex]. The quadratic formula is:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
4. Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the quadratic formula:
[tex]\[
x = \frac{{-(-16) \pm \sqrt{{(-16)^2 - 4 \cdot 1 \cdot 65}}}}{2 \cdot 1}
\][/tex]
5. Simplify the expression inside the square root (the discriminant):
[tex]\[
(-16)^2 = 256
\][/tex]
[tex]\[
4 \cdot 1 \cdot 65 = 260
\][/tex]
[tex]\[
256 - 260 = -4
\][/tex]
This means the discriminant is [tex]\(-4\)[/tex].
6. Since the discriminant is negative, the solutions to the quadratic equation will be complex numbers. Thus, we have:
[tex]\[
x = \frac{{16 \pm \sqrt{-4}}}{2}
\][/tex]
7. The square root of [tex]\(-4\)[/tex] is [tex]\(2i\)[/tex] ([tex]\(i\)[/tex] is the imaginary unit). Therefore:
[tex]\[
x = \frac{{16 \pm 2i}}{2}
\][/tex]
8. Simplify the expression:
[tex]\[
x = 8 \pm i
\][/tex]
So, the solutions for the quadratic equation [tex]\( x^2 = 16x - 65 \)[/tex] are:
[tex]\[
x_1 = 8 + i
\][/tex]
[tex]\[
x_2 = 8 - i
\][/tex]
