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The substitution method is a powerful tool in solving complex equations by simplifying them. When you encounter equations like the one given in the exercise, it is often beneficial to reduce the complexity by introducing a new variable. This is done by substituting a part of the equation with something simpler.

For instance, in the equation \(x^4 - 8x^2 + 2 = 0\), a direct solution could be cumbersome. Instead, we substitute \(y = x^2\), transforming the original equation into \(y^2 - 8y + 2 = 0\), which is a standard quadratic equation.

The idea is straightforward:

• Identify a term that can make the equation simpler.
• Substitute it with a new variable.
• Solve the new and simpler equation.

Once resolved, remember to backtrack and replace the substitution to find the actual solution for the original variable, \(x\), in this case.

The quadratic formula is a reliable method for solving quadratic equations, of the form \(ax^2 + bx + c = 0\). In our exercise, once the substitution is made, we apply the quadratic formula to the equation \(y^2 - 8y + 2 = 0\). This formula provides the solutions for \(y\) by calculating:

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

Here, \(a = 1\), \(b = -8\), and \(c = 2\). Plugging these into the formula allows us to solve for \(y\). It uses basic operations of addition, subtraction, multiplication, division, and finding a square root, making it universal and efficient.

This formula is essential for every student learning quadratic equations because it guarantees a method to find the solutions when other methods, like factoring, are either too complex or inapplicable.

The discriminant is a crucial component of the quadratic formula, as it helps determine the nature of the roots of a quadratic equation.
In the formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), the expression \(b^2 - 4ac\) inside the square root is the discriminant, which we denote as \(D\). In the exercise, for \(y^2 - 8y + 2 = 0\), the discriminant is calculated as:

• \(D = (-8)^2 - 4 \cdot 1 \cdot 2 = 64 - 8 = 56\)

The value of the discriminant tells us about the real and complex nature of the roots:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root (a repeated root).
• If \(D < 0\), the equation has complex roots.

In this case, since \(D = 56\), which is greater than zero, it confirms that the equation has two distinct real roots.