91Ó°ÊÓ

The substitution method is a handy tool in algebra for understanding complex polynomial equations. Often, you can simplify a difficult equation by substituting one variable for another, leading to a more manageable form. In this exercise, we simplified the given quartic equation, \(x^4 - 18x^2 + 72 = 0\), by using the substitution \(y = x^2\).
This transformation changes a complex equation into a simpler quadratic one, \(y^2 - 18y + 72 = 0\). Here, the substitution helped us reduce the problem from dealing with an equation in \(x^4\) to one in \(y^2\), which is easier to solve.

• Always identify if a substitution can simplify your equation.
• Select appropriate values for substitution that naturally fit the structure of the equation.
• Translate back the substituted values to find the solution to the original equation.

Substitution is particularly useful when dealing with powers of the variable that can be expressed as a product or power of a substituted term, like \(x^2\) in our case.

The quadratic formula is a powerful tool for finding solutions to quadratic equations of the form \(ax^2 + bx + c = 0\). In our transformed equation \(y^2 - 18y + 72 = 0\), it is used to find possible values of \(y\).
The formula itself is given by:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a\), \(b\), and \(c\) are coefficients from our quadratic equation. For it to work smoothly:

• Assign values to \(a\), \(b\), and \(c\) correctly. In this case, \(a = 1\), \(b = -18\), and \(c = 72\).
• Calculate the discriminant \(b^2 - 4ac\) to understand the nature of the roots.
• Solve the equation carefully, noting each plus and minus sign.

The quadratic formula is invaluable as it secures all potential roots, and it is essential for anyone dealing with quadratic problems.

The discriminant, \(b^2 - 4ac\), is a component of the quadratic formula that determines the nature of a quadratic's roots. In the formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), the discriminant lies under the square root.
In our scenario, it was calculated as:
\[(-18)^2 - 4 \times 1 \times 72 = 324 - 288 = 36\]
This tells us how many solutions exist and what type they are:

• If the discriminant is positive, like in this case (36), there are two distinct real roots.
• If it is zero, there is exactly one real repeated solution.
• If negative, the quadratic has no real solutions but two complex ones.

Understanding the discriminant is crucial when solving quadratics, as it provides insight into how the quadratic behaves and what kind of solution to expect.