Problem 56 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \frac{\sqrt{x^{2}-2 x-8}}{\sqrt{x+2}} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: $$\frac{\sqrt{x^{2} - 2x - 8}}{\sqrt{x + 2}}$$ Answer: $$\sqrt{x - 4}$$

Step 1: Factor the quadratic expression in the numerator

First, we need to factor the quadratic expression in the numerator: $$ x^2 - 2x - 8 $$ To do this, we look for two numbers that multiply to -8 and add to -2. The numbers -4 and +2 satisfy these conditions, so we can write the expression as: $$ x^2 - 2x - 8 = (x - 4)(x + 2) $$ Now we can rewrite the given expression as: $$ \frac{\sqrt{(x - 4)(x + 2)}}{\sqrt{x + 2}} $$

Step 2: Simplify the square roots

Next, we can simplify the expression by eliminating the square roots: $$ \frac{\sqrt{(x - 4)(x + 2)}}{\sqrt{x + 2}} = \sqrt{\frac{(x - 4)(x + 2)}{x + 2}} $$ Now we can see that the \((x + 2)\) term in the numerator and denominator cancel each other out: $$ \sqrt{\frac{(x - 4)(x + 2)}{x + 2}} = \sqrt{x - 4} $$

Step 3: Write the final answer

The simplified expression is: $$ \frac{\sqrt{x^{2} - 2x - 8}}{\sqrt{x + 2}} = \sqrt{x - 4} $$

Key Concepts

Factoring Quadratics

Factoring quadratics is an essential skill in algebra that helps in simplifying expressions and solving equations. A quadratic expression is typically in the form of \( ax^2 + bx + c \). The goal of factoring is to express it as a product of two binomials. In the exercise, we started with \( x^2 - 2x - 8 \). To factor this, we need to find two numbers that multiply to \(-8\) and add up to \(-2\).

• These numbers are \(-4\) and \(+2\).
• Thus, the expression \( x^2 - 2x - 8 \) can be factored into \( (x - 4)(x + 2) \).

Once factored, the expression becomes easier to work with, as seen in the following steps of the solution.

Square Roots

Square roots help in determining the value that, when multiplied by itself, gives the original number. In the context of simplifying expressions, square roots also apply to simplifying fractional expressions where a square term appears.In the exercise, once the quadratic was factored to \( (x - 4)(x + 2) \), we dealt with square roots:

• We had \( \frac{\sqrt{(x - 4)(x + 2)}}{\sqrt{x + 2}} \).

Using the property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \), the expression can be simplified further by canceling \( (x + 2) \) from the numerator and the denominator, giving us \( \sqrt{x - 4} \). This cancellation step is critical for simplification.

Rational Expressions

Rational expressions are fractions that have polynomials in the numerator and the denominator. Simplifying these expressions often involves factoring and canceling common terms. In the given problem, the initial expression \( \frac{\sqrt{x^2 - 2x - 8}}{\sqrt{x + 2}} \) is a rational expression with square roots involved. After factoring and simplifying, it becomes apparent how important it is to:

• Factor where possible to expose common terms.
• Simplify by canceling these common terms.

After simplifying the initial rational expression by factoring the quadratic in the numerator and then canceling \( (x + 2) \) from the numerator and denominator, we minimized the expression to its simplest form, \( \sqrt{x - 4} \). Understanding these steps is key to mastering rational expressions.