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Factoring is a simple and effective method used to solve quadratic equations by expressing them as a product of simpler expressions. To factor a quadratic equation like \( 3x^2 - 27 = 0 \), first look for a common factor in each term. Here, both \( 3x^2 \) and \(-27\) can be divided by 3, giving us:

• Factor out the greatest common factor: \( 3(x^2 - 9) = 0 \).
• Recognize the expression inside the parentheses, \( x^2 - 9 \), as a difference of squares.
• Applying the difference of squares formula, we get \((x + 3)(x - 3)\).

The factored equation is \( 3(x + 3)(x - 3) = 0 \). The zero product property tells us that if the product of factors equals zero, then at least one factor must be zero. Solving \( x + 3 = 0 \) and \( x - 3 = 0 \) gives us the solutions \( x = -3 \) and \( x = 3 \). Factoring can be a quick and neat method when the quadratic term is easily factorable like in this case.

The quadratic formula is a universal method for solving any quadratic equation, regardless of whether it can be easily factored or not. It is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation in standard form \( ax^2 + bx + c = 0 \).In our original problem, \( 3x^2 - 27 = 0 \), we identify \( a = 3 \), \( b = 0 \), and \( c = -27 \). Applying these values into the quadratic formula:

• Calculate the discriminant: \( b^2 - 4ac = 0^2 - 4(3)(-27) = 324 \).
• The formula becomes: \( x = \frac{0 \pm \sqrt{324}}{6} \).
• Simplifying gives us \( x = \frac{\pm 18}{6} \).

This yields the solutions \( x = 3 \) and \( x = -3 \). The quadratic formula is particularly useful when the quadratic equation is not easily factorable, providing a straightforward approach to finding roots.

The difference of squares is a specific factoring pattern that appears frequently in solving quadratic equations. It's written as:\[ a^2 - b^2 = (a + b)(a - b) \]This formula is extremely handy because it allows us to rewrite expressions in a multiplicative form which is often easier to solve.In our problem \( 3x^2 - 27 = 0 \), after factoring out 3, we observe \( x^2 - 9 \), which can be factored using this pattern:

• Recognize \( x^2 \) as \( a^2 \) where \( a = x \).
• Recognize \( 9 \) as \( b^2 \) where \( b = 3 \).
• Apply the difference of squares: \( x^2 - 9 = (x + 3)(x - 3) \).

Using the difference of squares simplifies the equation into a product of two binomials, \( (x + 3)(x - 3) \), allowing for easy application of the zero product property to solve for \( x \). Understanding the difference of squares is an important algebraic technique for quickly finding solutions in quadratic equations that suit this pattern.