To determine if the equation [tex]\( x^9 - 5x^3 + 6 = 0 \)[/tex] is quadratic in form, let's analyze and break down its structure step by step.
1. Understand the Concept of Quadratic in Form:
A quadratic in form means the equation can be restructured or rewritten in the standard quadratic equation format:
[tex]\[
(some \ function \ of \ x)^2 + (some \ function \ of \ x) + constant = 0
\][/tex]
2. Substitution for Analysis:
Let's introduce a substitution to simplify and analyze the given equation. Suppose [tex]\( y = x^3 \)[/tex]. This substitution helps reframe higher powers of [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
3. Transform the Equation:
Using the substitution [tex]\( y = x^3 \)[/tex], we note that:
[tex]\[
y^3 = (x^3)^3 = x^9
\][/tex]
Substitute [tex]\( y \)[/tex] into the original equation:
[tex]\[
x^9 - 5x^3 + 6 = 0
\][/tex]
This becomes:
[tex]\[
y^3 - 5y + 6 = 0
\][/tex]
4. Examine the Transformed Equation:
Now, we have the equation [tex]\( y^3 - 5y + 6 = 0 \)[/tex]. Observe the form of this equation.
- In the standard quadratic form [tex]\( ay^2 + by + c = 0 \)[/tex], the highest power of [tex]\( y \)[/tex] should be 2. Here, we have [tex]\( y^3 \)[/tex].
- Since [tex]\( y^3 - 5y + 6 = 0 \)[/tex] has the highest power of 3, it is actually a cubic equation in terms of [tex]\( y \)[/tex].
5. Conclusion:
Because we cannot rewrite the transformed equation in the standard quadratic form [tex]\( ay^2 + by + c = 0 \)[/tex] (it is cubic instead), we conclude that the original equation [tex]\( x^9 - 5x^3 + 6 = 0 \)[/tex] is not quadratic in form.
Therefore, the equation [tex]\( x^9 - 5x^3 + 6 = 0 \)[/tex] is not quadratic in form because it cannot be expressed as [tex]\( (some \ function \ of \ x)^2 + (some \ function \ of \ x) + \text{constant} = 0 \)[/tex]. Instead, it is a cubic equation in terms of [tex]\( y = x^3 \)[/tex].
