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The factoring method is a powerful tool in solving quadratic equations. It involves rewriting the quadratic expression as a product of two binomial expressions. For a quadratic equation of the form:

• \( ax^2 + bx + c = 0 \)

we find two numbers that multiply to the constant term \( c \) and add to the linear coefficient \( b \). This step helps in splitting the middle term and rewriting the quadratic as a factored expression.

In our example given by the equation \( x^2 + 4x - 21 = 0 \), we need numbers that multiply to -21 and add to 4. The numbers 7 and -3 work because:

• \( 7 \times (-3) = -21 \)
• \( 7 + (-3) = 4 \)

This identification allows the middle term \( 4x \) to be split into \( 7x - 3x \), so that the quadratic can be rewritten in a way that reveals its factorable components.

Quadratic equations can be solved via several methods, one of which is factoring, as demonstrated in this exercise. Solving involves finding the values of \( x \) that make the equation equal zero. After factoring the quadratic equation, the next step:

• Express the quadratic in the form \((x - p)(x - q) = 0\)

The solutions to the equation are

• \( x = p \)
• \( x = q \)

For the equation \( (x + 7)(x - 3) = 0 \), it leads to:

• \( x + 7 = 0 \Rightarrow x = -7 \)
• \( x - 3 = 0 \Rightarrow x = 3 \)

These solutions are then verified by substituting back into the original equation to ensure they satisfy the quadratic.

To efficiently tackle quadratic equations, certain algebraic techniques are essential. Understanding the concept of factoring, grouping, and solving is key.

Let's take a closer look at the original equation:

• Group terms for ease of factorization, i.e., \( (x^2 + 7x) + (-3x - 21) \).
• Extract common factors from each group, resulting in \( x(x + 7) - 3(x + 7) \).
• Notice the repeated binomial \( (x + 7) \), which is a common factor.

By factoring the common binomial, the expression simplifies, leading us to the solutions.

This use of algebraic techniques not only aids in solving the equation but also builds foundational skills for more complex algebraic manipulations.