Understanding the Quartic Equation

We are given the equation $x^4 - 13x^2 + 36 = 0$. This is a quartic equation because the highest power of the variable $x$ is 4. However, notice that the equation only contains terms with even powers of $x$ (namely $x^4$ and $x^2$) and a constant term. This structure allows us to simplify it into a quadratic equation.

Reducing the Equation to Quadratic Form

To reduce the equation, we can use a substitution. Let $y = x^2$. Since $x^4 = (x^2)^2$, we can rewrite the equation in terms of $y$:

Substituting $y$ for $x^2$: $$(y)^2 - 13(y) + 36 = 0$$

This simplifies to the quadratic equation:

This is the reduced quadratic equation we needed to find.

Finding Roots of the Reduced Quadratic Equation

Now, we need to find the roots of the quadratic equation $y^2 - 13y + 36 = 0$. We can solve this by factoring. We are looking for two numbers that multiply to 36 and add up to -13.

Let's list factors of 36:

The numbers -4 and -9 satisfy both conditions. Therefore, we can factor the quadratic equation as:

To find the roots, we set each factor equal to zero:

1. $y - 4 = 0 \implies y = 4$

2. $y - 9 = 0 \implies y = 9$

So, the roots of the reduced quadratic equation $y^2 - 13y + 36 = 0$ are $y = 4$ and $y = 9$. These are the values requested by the question.