Problem 5 - Algebra 2

Question

Identify all values of \(y\) for which \(\frac{y-4}{y^{2}-4 y-12}\) is undefined. A. -2, 4, 6 B. -6, -4, 2 C. -2, 0, 6 D. -2, 6

Step-by-Step Solution

Answer

The expression is undefined at \(y = -2\) and \(y = 6\) (option D).

Step 1: Understand the Expression

The expression given is \(\frac{y-4}{y^2 - 4y - 12}\). This expression is undefined wherever the denominator equals zero.

Step 2: Set the Denominator to Zero

We need to solve the equation \(y^2 - 4y - 12 = 0\) to find the values of \(y\) where the denominator is zero, as these are the values that make the expression undefined.

Step 3: Factor the Quadratic Equation

Rewrite the quadratic equation \(y^2 - 4y - 12 = 0\) in factored form. This requires finding two numbers whose product is \(-12\) and sum is \(-4\). The numbers are \(2\) and \(-6\). So, the factored form is \((y - 6)(y + 2) = 0\).

Step 4: Solve for Values of y

Set each factor equal to zero to solve for \(y\).\(y - 6 = 0 \Rightarrow y = 6\)\(y + 2 = 0 \Rightarrow y = -2\)These are the values of \(y\) that make the denominator zero.

Step 5: Conclude the Solution

The expression \(\frac{y-4}{y^2 - 4y - 12}\) is undefined for \(y = 6\) and \(y = -2\), which matches the values in option D.

Key Concepts

Undefined Expressions

Rational expressions become undefined whenever their denominator equals zero. This is because division by zero is not allowed in mathematics. In the expression \( \frac{y-4}{y^2 - 4y - 12} \), we're interested in finding the values of \( y \) that make this denominator equal to zero. To do this, we take the denominator \( y^2 - 4y - 12 \) and set it equal to zero: \( y^2 - 4y - 12 = 0 \). Solving this equation will provide the values of \( y \) that will result in the expression being undefined. These values are important to identify, as they form the restrictions on the domain of the rational expression.

Quadratic Equations

A quadratic equation is any equation that can be rewritten in the standard form \( ax^2 + bx + c = 0 \). For the expression \( \frac{y-4}{y^2 - 4y - 12} \), the denominator \( y^2 - 4y - 12 \) is a quadratic equation. Quadratic equations often have two solutions, which can be found using various methods such as factoring, completing the square, or using the quadratic formula. In this context, solving the quadratic equation is essential for determining when the rational expression is undefined, as it helps pinpoint the values that zero out the denominator.

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic equation as a product of two binomials. For \( y^2 - 4y - 12 = 0 \) to make it easier to identify values that make the expression undefined, we need to factor it. We look for two numbers that multiply to \(-12\) and add up to \(-4\). These numbers are \(2\) and \(-6\). Thus, we rewrite the equation as \((y - 6)(y + 2) = 0 \). This factoring process simplifies solving quadratic equations by breaking them down into simpler terms, revealing the values that cause the denominator to be zero and making the rational expression undefined.

Solving Equations

Once a quadratic equation is factored, solving it involves setting each factor separately to zero. In the equation \((y - 6)(y + 2) = 0\), we set each binomial to zero:

• For \( y - 6 = 0 \), solve by adding \(6\) to both sides, yielding \( y = 6 \).
• For \( y + 2 = 0 \), solve by subtracting \(2\) from both sides, yielding \( y = -2 \).

These solutions, \( y = 6 \) and \( y = -2 \), are the values that make the original rational expression undefined. Solving the factored equation pinpoint precisely when the expression loses its validity and clearly indicates the restrictions on its domain.