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When faced with higher-degree polynomials, such as a fourth-degree polynomial, it can be challenging to find the roots directly. In such cases, using a quadratic substitution is a helpful technique. It involves substituting a variable with another quadratic expression, simplifying the equation, and then solving it as a standard quadratic equation.

In the problem given, the original function is a fourth-degree polynomial:

• Polynomial: \(g(x)=x^{4}-13 x^{2}+36\)
• Notice: It contains terms like \(x^4\) and \(x^2\)

To simplify, recognize that setting \(y = x^2\) transforms the polynomial into a quadratic in terms of \(y\). So, the equation becomes:

• Substitute \(y=x^{2}\): \(g(y) = y^2 - 13y + 36\)

This conversion makes it easier to manage and solve, as you now have a familiar quadratic form. After finding the solutions for \(y\), remember to convert back to \(x\) using the original substitution.

The quadratic formula is an essential tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It provides a straightforward method to calculate the roots directly, regardless of whether they are real or complex.

Here is the quadratic formula:

• Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Using the example from the problem, we substitute \(a=1\), \(b=-13\), and \(c=36\) for the quadratic \(g(y) = y^2 - 13y + 36\).

Applying the quadratic formula:

• Calculate roots: \(y = \frac{-(-13) \pm \sqrt{(-13)^2 - 4\cdot 1\cdot 36}}{2\cdot 1}\)
• Calculate: \(y = \frac{13 \pm \sqrt{169 - 144}}{2}\)
• Simplify further: \(y = \frac{13 \pm \sqrt{25}}{2}\)

This results in solutions for \(y = 9\) and \(y = 4\). After finding these values, re-substitute \(y = x^2\) to continue solving for \(x\).

The discriminant in the quadratic formula is the expression beneath the square root, \(b^2 - 4ac\). It provides valuable information about the nature of the roots of the quadratic equation.

Understanding the discriminant:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root (a double root).
• If \(b^2 - 4ac < 0\), the roots are complex or imaginary.

In the given problem, calculate the discriminant for \(y^2 - 13y + 36\) using:

• Calculate: \(169 - 144 = 25\)

Since 25 is greater than zero, it confirms that there are two distinct real roots for \(y\). Knowing the discriminant helps in predicting the type of roots without fully solving the equation, making it a powerful preliminary check.