To solve the equation [tex]\( x^4 - 17x^2 + 16 = 0 \)[/tex], we can use a substitution to simplify the problem.
1. Substitute:
Let [tex]\( u = x^2 \)[/tex].
This means [tex]\( u^2 = (x^2)^2 = x^4 \)[/tex].
2. Rewrite the equation:
Substitute [tex]\( u \)[/tex] into the original equation:
[tex]\[
x^4 - 17x^2 + 16 = 0 \implies u^2 - 17u + 16 = 0
\][/tex]
Now, we have a quadratic equation in terms of [tex]\( u \)[/tex].
3. Solve the quadratic equation:
To solve [tex]\( u^2 - 17u + 16 = 0 \)[/tex], we can use the quadratic formula [tex]\( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -17 \)[/tex], and [tex]\( c = 16 \)[/tex].
- Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = (-17)^2 - 4 \cdot 1 \cdot 16 = 289 - 64 = 225
\][/tex]
- Find the solutions for [tex]\( u \)[/tex]:
[tex]\[
u_1 = \frac{17 + \sqrt{225}}{2 \cdot 1} = \frac{17 + 15}{2} = \frac{32}{2} = 16
\][/tex]
[tex]\[
u_2 = \frac{17 - \sqrt{225}}{2 \cdot 1} = \frac{17 - 15}{2} = \frac{2}{2} = 1
\][/tex]
4. Back-substitute to find [tex]\( x \)[/tex]:
Since [tex]\( u = x^2 \)[/tex], we need to solve for [tex]\( x \)[/tex]:
- For [tex]\( u_1 = 16 \)[/tex]:
[tex]\[
x^2 = 16 \implies x = \pm \sqrt{16} = \pm 4
\][/tex]
- For [tex]\( u_2 = 1 \)[/tex]:
[tex]\[
x^2 = 1 \implies x = \pm \sqrt{1} = \pm 1
\][/tex]
5. List all solutions:
The solutions to the original equation [tex]\( x^4 - 17x^2 + 16 = 0 \)[/tex] are:
[tex]\[
x = 4, -4, 1, -1
\][/tex]
So, the complete set of solutions is:
[tex]\[
x = \{4, -4, 1, -1\}
\][/tex]
