Question

How do you convert depressed quartic to a quadratic?

Answer 1

Hint: In this problem, we have to convert a depressed quartic equation into a quadratic. To do this, we should know that the quartic equations are polynomials that have degrees of four where quadratic equations are polynomials that have degrees of two. We can first take a depressed quartic equation by setting it equal to zero. We can then replace the terms with exponent to another variable to get a quadratic equation.

Complete step by step answer:
We know that the quartic equations are polynomials that have degrees of four where quadratic equations are polynomials that have degrees of two.
We can now take a depressed quartic equation.
\[{{x}^{4}}-81=0\]
We can see that it has two terms, a fourth power and a constant.
We can now replace the term \[{{x}^{2}}\] with variable r.
We know that this means \[{{x}^{2}}=r\], by replacing it, we get
\[\Rightarrow {{r}^{2}}-81=0\]
Therefore, the quartic equation is converted into a quartic equation.
 We can now take another example of a depressed quartic equation.
\[6{{x}^{4}}-35{{x}^{2}}+50=0\]
We can see that it has three terms, a fourth power, a square and a constant.
We can now replace the term \[{{x}^{2}}\] with variable r.
We know that this means \[{{x}^{2}}=r\], by replacing it, we get
\[\Rightarrow 6{{r}^{2}}-35r+50=0\]
Therefore, the quartic equation is converted into a quartic equation.

Note:
We should know that sometimes a quartic equation can look like a quadratic equation and have three terms. If a quartic has a term raised to the fourth power, a term raised to a second power and a constant, we can substitute \[{{x}^{2}}\] with any other variable and then we can treat it like a quadratic equation.