Problem 98 - Elementary Algebra

Question

For Problems \(93-106\), evaluate each algebraic expression for the given values of the variables. $$ x^{2}+2 x y+y^{2} \quad \text { for } x=-\frac{3}{2} \text { and } y=-2 $$

Step-by-Step Solution

Answer

The evaluated expression is \( \frac{49}{4} \).

Step 1: Substitute the Given Values

We start with the expression \( x^{2}+2xy+y^{2} \). The values given are \( x = -\frac{3}{2} \) and \( y = -2 \). Substitute these values into the expression, which becomes \( \left(-\frac{3}{2}\right)^{2} + 2\left(-\frac{3}{2}\right)(-2) + (-2)^{2} \).

Step 2: Calculate \( x^2 \)

Compute \( \left(-\frac{3}{2}\right)^2 \). This equals \( \frac{9}{4} \) since squaring a negative number results in a positive number.

Step 3: Calculate \( y^2 \)

Compute \( (-2)^2 \). This equals 4, as squaring a negative number results in a positive number.

Step 4: Calculate \( 2xy \)

Compute \( 2 \times -\frac{3}{2} \times -2 \). First, multiply \( 2 \times -\frac{3}{2} \), which equals \(-3\). Then, \(-3 \times -2 = 6\).

Step 5: Sum the Calculated Values

Add up \( \frac{9}{4} \), 6, and 4. First, convert 6 into a fraction with the same denominator: \( 6 = \frac{24}{4} \), and 4 as \( 4 = \frac{16}{4} \). Now, sum them up: \( \frac{9}{4} + \frac{24}{4} + \frac{16}{4} = \frac{49}{4} \).

Step 6: Simplify the Expression

The sum \( \frac{49}{4} \) does not simplify further, as 49 and 4 do not have a common factor. Therefore, the simplified expression remains \( \frac{49}{4} \).

Key Concepts

Substitution Method

The substitution method in algebraic expressions is pivotal when you need to evaluate specific variable values. It involves replacing each variable in an expression with the corresponding numerical value.
Start with the given expression and carefully substitute all the variables with the numbers provided in the problem. For instance:

• Given expression: \( x^{2}+2xy+y^{2} \)
• Substitute: \( x = -\frac{3}{2} \) and \( y = -2 \)
• Expression becomes: \[ \left(-\frac{3}{2}\right)^2 + 2\left(-\frac{3}{2}\right)(-2) + (-2)^2 \]

This approach helps keep track of calculations step-by-step, ensuring correct results.

Exponents

Understanding exponents is crucial when dealing with algebraic expressions. An exponent indicates how many times to multiply a number by itself.
For example:

• \( x^2 \) means \( x \) multiplied by itself.
• So, \( (-\frac{3}{2})^2 = \left(-\frac{3}{2}\right) \times \left(-\frac{3}{2}\right) = \frac{9}{4} \).

Remember:

• A negative number squared results in a positive product.
• To simplify calculations, handle fractions carefully to avoid errors.

Fractions

Fractions play an essential role in algebra and require careful handling to compute expressions correctly.
When multiplying fractions, multiply the numerators and denominators separately. For subtraction or addition, consider a common denominator.

• Multiply: \( 2 \times -\frac{3}{2} \) becomes \(-3\) since \( 2 \cdot \frac{1}{2} = 1 \).
• Add: Convert integers such as 6 into fractions like \( \frac{24}{4} \) to combine easily with \( \frac{9}{4} \).

Be meticulous with fraction simplifications and ensure everything retains a consistent format for successful calculation.

Polynomial Evaluation

Evaluating polynomials involves substituting given values and simplifying the expression step by step.
In this problem, you start by inserting the specific values and carry through calculations for each term of the polynomial:

• Evaluate\( x^2 \) and \( y^2 \) separately.
• Calculate and simplify each portion before combining.
• The expression resolves to \( \frac{49}{4} \) after successful computation of each part: \( \frac{9}{4} + 6 + 4 \).

Breaking it down maintains clarity, allowing focus on how each term contributes to the final result.