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Quadratic equations are polynomial equations of degree two. They typically appear in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The highest power of the variable \(x\) is two, which is why these equations are called 'quadratic'—from the Latin word 'quadratus' meaning "square."
Quadratic equations are vital in algebra because they model various real-world situations, like the path of a projectile or area problems. Solving these equations can seem tricky, but various methods help achieve the solution, such as factoring, completing the square, and using the quadratic formula.
Understanding the structure of quadratic equations and identifying the coefficients and constant terms helps in deploying the right strategy for solving them. Each method offers insights into the properties of parabolas—the graphs of quadratic equations.

Factoring is a straight-forward method for solving quadratic equations. It involves expressing the quadratic equation as a product of its linear factors. This method works well when the quadratic can be easily decomposed into two binomials.
Here's a simple guide to factoring:

• Set the quadratic equation to zero.
• Find two numbers that multiply to the constant term \(c\) and add up to the coefficient \(b\).
• Express the quadratic as \((x + m)(x + n)\), where \(m\) and \(n\) are the numbers found.

In the given example, we started with \(x^2 + 9x - 112\). By recognizing that 16 and -7 are the numbers satisfying the conditions, we could write the quadratic as \((x + 16)(x - 7)\). Factoring is intuitive once you recognize patterns in various quadratics.

Algebraic expressions are combinations of numbers, variables, and operators used in equations. They form the foundation of algebra and are used to express mathematical relationships.
In a quadratic equation like \(x^2 + 9x - 112\), the terms consist of a squared variable term, a linear variable term, and a constant term. Understanding these components is vital for any manipulation, such as factoring.
Algebraic expressions can be manipulated using arithmetic operations, distribution, and substitution. Learning to handle expressions is crucial because it sets the stage for solving equations and inequalities across mathematics. Simplifying algebraic expressions, like separating a trinomial into binomials, helps uncover the values that satisfy the original equation.