Sure, let's solve the equation [tex]\( x^4 - 17x^2 + 16 = 0 \)[/tex] step by step.
1. Substitute [tex]\( u = x^2 \)[/tex]:
By substituting [tex]\( u = x^2 \)[/tex], we can rewrite the equation in terms of [tex]\( u \)[/tex]:
[tex]\[
x^4 - 17x^2 + 16 = 0 \implies u^2 - 17u + 16 = 0
\][/tex]
This substitution simplifies the quartic equation into a quadratic equation.
2. Factor the quadratic equation:
We need to factor [tex]\( u^2 - 17u + 16 = 0 \)[/tex]:
[tex]\[
u^2 - 17u + 16 = (u - 16)(u - 1) = 0
\][/tex]
This factorization shows that the solutions for [tex]\( u \)[/tex] are [tex]\( u = 16 \)[/tex] and [tex]\( u = 1 \)[/tex].
3. Solve for [tex]\( x \)[/tex] by substituting back [tex]\( u = x^2 \)[/tex]:
- For [tex]\( u = 16 \)[/tex]:
[tex]\[
x^2 = 16 \implies x = \pm 4
\][/tex]
- For [tex]\( u = 1 \)[/tex]:
[tex]\[
x^2 = 1 \implies x = \pm 1
\][/tex]
4. Collect all solutions:
The complete set of solutions for [tex]\( x \)[/tex] is:
[tex]\[
x = 4, x = -4, x = 1, x = -1
\][/tex]
So, the solutions to the original equation [tex]\( x^4 - 17x^2 + 16 = 0 \)[/tex] are:
[tex]\[
x = 4, x = -4, x = 1, x = -1
\][/tex]
