Problem 80 - Elementary Algebra

Question

For exercises 77-86, find any values of the variable for which this expression is undefined. $$ \frac{y^{2}+4 y}{y^{2}-8 y-20} $$

Step-by-Step Solution

Answer

The expression is undefined for \(y = 10\) and \(y = -2\).

Step 1: Identify the denominator

Focus on the denominator of the given expression: \(y^2 - 8y - 20\).

Step 2: Set the denominator equal to zero

An expression is undefined when the denominator equals zero. Set up the equation: \(y^2 - 8y - 20 = 0\).

Step 3: Factor the quadratic equation

Factor the quadratic equation to solve for the values of \(y\). You are looking for two numbers that multiply to \(-20\) and add to \(-8\): \(y^2 - 8y - 20 = (y - 10)(y + 2) = 0\).

Step 4: Solve for y

Set each factor equal to zero and solve for \(y\): \(y - 10 = 0\) and \(y + 2 = 0\).\(y = 10\) and \(y = -2\).

Step 5: State the final answer

The expression is undefined for the values of \(y = 10\) and \(y = -2\).

Key Concepts

Quadratic Equations

A quadratic equation is any equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. In our exercise, we are dealing with the quadratic equation y^2 - 8y - 20 = 0. Quadratic equations can be solved by various methods such as factoring, completing the square, and using the quadratic formula. They are fundamental in algebra because they appear in various contexts ranging from physics to finance. The solutions to a quadratic equation are the values of y that satisfy the equation, and they are also known as the roots of the equation.

Factoring

Factoring is a method used to simplify expressions or solve equations by rewriting them as a product of simpler expressions. In the context of quadratics, factoring involves writing the quadratic expression as a product of two binomials. For example, in the step from the problem, y^2 - 8y - 20 can be factored into (y - 10)(y + 2). Factoring requires finding two numbers that not only multiply to give the constant term (-20 in this case) but also add to give the coefficient of the linear term (-8). When factoring is complete, setting each factor equal to zero gives the solutions of the equation.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. In our problem, we have the rational expression \(\frac{y^2+4 y}{y^2-8 y-20}\). Working with rational expressions often involves simplifying them by factoring both the numerator and the denominator and canceling common factors. When simplifying rational expressions, it's crucial to find the values that make the denominator zero because those values make the expression undefined and thus are excluded from the solution set.

Undefined Values

Undefined values in algebraic expressions are those that make the denominator zero, leading to division by zero, which is undefined in mathematics. In our exercise, to find the undefined values, we set the denominator equal to zero: y^2 - 8y - 20 = 0. By solving this equation, we find the values of y that make the denominator zero, which are y = 10 and y = -2. Thus, the original rational expression \(\frac{y^2+4 y}{y^2-8 y-20}\) is undefined for y = 10 and y = -2. Identifying these values is essential for understanding the domain of rational expressions.