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A quadratic equation is any equation that takes the form \( ax^2 + bx + c = 0 \). To solve quadratic equations, we look for values of \( x \) that make the equation equal to zero. These values are called the 'zeros' or 'roots' of the equation.

This type of equation is called 'quadratic' because it involves \( x^2 \), which is a term where the variable is raised to the second power. When solving quadratic equations, we typically use methods like factoring, completing the square, or the quadratic formula.

In our example, the quadratic function given is \( f(x) = x^2 - x - 2 \). The goal is to find the zeros of this function, i.e., the values of \( x \) for which \( f(x) = 0 \). This leads us to solve the equation \( x^2 - x - 2 = 0 \).

Factoring is a method used to simplify polynomials by breaking them down into 'factors'—simpler expressions that, when multiplied together, give back the original polynomial.

For quadratic equations, factoring often involves finding two binomials (expressions containing two terms) that when multiplied together produce the original quadratic expression.

In the given exercise, we need to factor \( x^2 - x - 2 \). We look for two numbers that multiply to \(-2\) (the constant term) and add up to \(-1\) (the coefficient of the middle term). These numbers are \(-2\) and \(1\). Therefore, we can write: \[ x^2 - x - 2 = (x - 2)(x + 1) \]

Factoring transforms the quadratic equation into a product of two binomials, making it easier to find its roots.

Algebraic solutions involve using algebraic techniques to find the roots or zeros of an equation. Once we have factored the quadratic expression into binomials, we set each factor equal to zero and solve for \( x \). This process is based on the 'zero-product property,' which states that if the product of two factors is zero, at least one of the factors must be zero.

Following our exercise, after factoring \( x^2 - x - 2 \) into \( (x - 2)(x + 1) \), we set each factor to zero:

• \( x - 2 = 0 \)
• \( x + 1 = 0 \)

We then solve these equations to find the zeros:

• For \( x - 2 = 0 \), solving for \( x \) gives \( x = 2 \)
• For \( x + 1 = 0 \), solving for \( x \) gives \( x = -1 \)

The roots of the quadratic equation \( x^2 - x - 2 = 0 \) are \( x = 2 \) and \( x = -1 \). These are the zeros of the function \( f(x) \).