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Factoring equations is a fundamental technique in algebra. It involves expressing a polynomial as a product of its factors. This method is essential for simplifying expressions and solving equations.
To factor an equation, you need to follow these steps:

• Identify the polynomial you want to factor
• Find values that, when multiplied, give you the original polynomial
• Express these values as a product of binomials or other simpler polynomials

An example of factoring includes breaking down a quadratic equation like the one in the exercise:
* Given: \(x^{2} - 144\)
* Recognize that \(144\) is a perfect square, specifically \(12^{2}\)
* The factored form would be \((x - 12)(x + 12)\).
This process makes it easier to solve the equation because it simplifies the polynomial into manageable parts.

A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\). To solve such equations, there are several methods, but one of the most powerful is factoring.
When you factor a quadratic equation, you aim to rewrite it in the form \((px + q)(rx + s) = 0\). This allows you to use the zero-factor property, which states that if the product of two factors is zero, one or both of the factors must be zero.
For instance, when given \(x^2 - 144 = 0\), and factored as \((x - 12)(x + 12) = 0\):

• You set each factor to zero: \(x - 12 = 0\) and \(x + 12 = 0\)
• Solving these gives you \(x = 12\) and \(x = -12\)

Hence, the solutions to the quadratic equation \(x^2 = 144\) are \(x = 12\) and \(x = -12\). These solutions are the points where the graph of the equation would intersect the x-axis, also known as the roots of the equation.

A perfect square is a number that can be expressed as the square of an integer. In simpler terms, if you can find an integer 'n' such that \(n^2\) equals the given number, then that number is a perfect square.
For example:

• 144 is a perfect square because \(12^2 = 144\)
• Similarly, \(25\) is a perfect square because \(5^2 = 25\)

Recognizing perfect squares is very helpful in algebra, especially when factoring quadratic equations. In our exercise, recognizing that 144 is a perfect square allowed us to write \(x^2 - 144\) as \((x - 12)(x + 12)\). This step was crucial in simplifying and solving the quadratic equation.
Understanding and identifying perfect squares can greatly simplify your work with polynomials and make solving quadratic equations much quicker and easier.