91Ó°ÊÓ

Quadratic equations are essential parts of algebra and appear frequently in mathematics. A standard quadratic equation has the form \(at^2 + bt + c = 0\), where \(a\), \(b\), and \(c\) are constants. The solutions to these equations, called roots, can be found using various methods, one of which is the quadratic formula:

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

For our problem, we form a quadratic equation using the given values \(x+y=12\) and \(xy=35\), which leads to \(t^2 - 12t + 35 = 0\). By substituting these into the quadratic formula, we can find the roots that represent the values of \(x\) and \(y\). This approach allows us to systematically identify the individual numbers that satisfy both given conditions. Using the roots, different expressions or functions can then be evaluated further.

The difference of squares is a powerful algebraic identity that simplifies the process of solving equations like \(x^2 - y^2\). It can be expressed as:

• \(x^2 - y^2 = (x+y)(x-y)\).

This factorization reveals a way to break down complicated expressions into simpler multiplicative components. Essentially, when you see a subtraction between two squared terms, you can quickly rewrite it in this factorized form. In the exercise, this identity helps us because we know \(x+y = 12\) and calculate \(x-y\).
It allows us to substitute directly into

• \((x+y)(x-y)\)

without needing to first determine \(x^2\) and \(y^2\) separately, which vastly simplifies the calculation process.

Simplifying expressions is a crucial skill in algebra, allowing one to reduce complex expressions into more manageable forms. In the given exercise, once we've recognized the identity \(x^2 - y^2 = (x+y)(x-y)\), we substitute known values into this simple expression.
From the steps, we know:

• \(x+y=12\)
• \(x-y=2\)

Substituting these into the identity, the expression simplifies swiftly:

• Calculate: \((x+y)(x-y) = 12 \times 2 = 24\).

This shows that simplifying expressions using identities not only aids in finding solutions faster but also enhances clarity. Instead of dealing with two distinct quadratic terms, we solve significantly easier multiplicative problems. Such techniques are essential for efficient problem-solving in mathematics and help minimize chances of error by reducing complex tasks.