Problem 40 - Intermediate Algebra

Question

Evaluate the algebraic expressions for the given values of the variables. $$ -x^{2}+2 x y+3 y^{2}, \quad x=-3 \text { and } y=3 $$

Step-by-Step Solution

Answer

The expression evaluates to 0.

Step 1: Substitute Values into Expression

Start by substituting the given values of the variables into the expression. The expression is \(-x^{2} + 2xy + 3y^{2}\). Substitute \(x = -3\) and \(y = 3\) into this expression to get: \[-(-3)^{2} + 2(-3)(3) + 3(3)^{2}\].

Step 2: Calculate Each Term Separately

Let's compute each term separately:1. \(-(-3)^{2}\): Compute \(-3\) squared: \((-3)^{2} = 9\). Thus, \(-9\).2. \(2(-3)(3)\): Compute: \(-3 \times 3 = -9\). Hence, \(2 \times (-9) = -18\). 3. \(3(3)^{2}\): Compute: \((3)^{2} = 9\). Thus, \(3 \times 9 = 27\).

Step 3: Sum the Results of Each Term

Add the results of each term from Step 2:1. \(-9\).2. \(-18\).3. \(+27\).Hence, \(-9 - 18 + 27 = 0\).

Step 4: Conclusion

The value of the expression when \(x = -3\) and \(y = 3\) is \(0\).

Key Concepts

Substitution

Substitution is a key process in evaluating algebraic expressions. When we have an expression with variables, like \(-x^2 + 2xy + 3y^2\), we replace the variables with actual numbers to compute the value of the expression. In this case, we're given the values \(x = -3\) and \(y = 3\), which are plugged directly into the expression.
By substituting, we transform the abstract algebraic idea into something concrete that can be solved with arithmetic. Think of substitution as translating a mathematical sentence into one that involves numbers directly.

• Identify variables in the expression: Here, they are \(x\) and \(y\).
• Replace each variable with the provided number: \(x = -3\) and \(y = 3\).
• The expression becomes numerical: \(-(-3)^2 + 2(-3)(3) + 3(3)^2\)

This step prepares our expression for the computational process.

Polynomial Simplification

Polynomial simplification is crucial when dealing with expressions that involve variables raised to powers. After substitution, we must simplify each part of the polynomial directly. Each term in the expression such as \(-x^2\), \(2xy\), and \(3y^2\) involves its own computation once the values are substituted.
Here's how you simplify:

• Determine the power applied to the variable, such as squaring in \((-3)^2\), which gives \(9\).
• Apply any negatives or multipliers to the result of that power. For \(-(-3)^2\), you apply the negative sign to \(9\), giving \(-9\).
• For each term, do the arithmetic explicitly: \(2(-3)(3) = -18\) and \(3(3)^2 = 27\).

By simplifying each term, you convert the expression into a series of simple arithmetic problems that can be easily added together at the end.

Arithmetic Operations

The final step is to add or subtract the simplified terms, involving basic arithmetic operations. These operations finalize the simplification process and provide the answer. We have the list of results from previous polynomial simplification: \(-9\), \(-18\), and \(27\). Now, they need to be summed.
Performing arithmetic operations on these values is straightforward:

• Start with the first number: \(-9\).
• Add \(-18\) to \(-9\): \(-9 - 18 = -27\).
• Add \(27\) to \(-27\): \(-27 + 27 = 0\).

Arithmetic operations involve simple adding or subtracting numbers, finalizing the calculation and giving the result of the whole expression. This neat closure means you're finished solving, arriving confidently at the number \(0\) for this particular exercise.