Problem 51 - Fundamentals of Mathematics

Question

Find the value of each expression. $$(x+8)^{2}+4(x+9)+1, \text { if } x=-6$$

Step-by-Step Solution

Answer

The value of the expression is 17.

Step 1: Substitute the value of x

Insert the given value of \( x = -6 \) into the expression \((x + 8)^2 + 4(x + 9) + 1\). This gives us:\[ ((-6) + 8)^2 + 4((-6) + 9) + 1 \]

Step 2: Simplify the parentheses

Calculate the expressions inside each parentheses:\((-6) + 8 = 2\) and \((-6) + 9 = 3\).Substitute these back into the expression:\[ (2)^2 + 4(3) + 1 \]

Step 3: Calculate the square of the number

Compute the square of the number obtained from the first parentheses:\((2)^2 = 4\).Substitute this back into the equation:\[ 4 + 4(3) + 1 \]

Step 4: Multiply the values

Multiply the numbers:\(4 \times 3 = 12\).Now replace the multiplication result in the expression:\[ 4 + 12 + 1 \]

Step 5: Add the numbers

Add the results obtained:\(4 + 12 = 16\) and then add 1.Finally, calculate:\[ 16 + 1 = 17 \]

Key Concepts

Substitution method

The substitution method is a powerful tool in algebra. It involves replacing variables with specific numbers to simplify an expression or find a particular solution. In our original exercise, we use substitution by taking the given value of \( x = -6 \) and placing it into the expression \((x + 8)^2 + 4(x + 9) + 1\). This transforms our original problem into a numerical expression:

• Replace \(x\) in \((x + 8)\) with \(-6\). This gives \((-6 + 8)\), which simplifies to \(2\).
• Replace \(x\) in \((x + 9)\) with \(-6\). This gives \((-6 + 9)\), which simplifies to \(3\).

This method allows us to work with numbers instead of variables, making the expression easier to handle as you move on to further calculations.

Simplification of expressions

Simplification is the process of reducing an algebraic expression to its simplest form. This often involves performing arithmetic operations and combining like terms. After using the substitution method, our expression becomes \((2)^2 + 4(3) + 1\).

Let's break down this simplification step-by-step:

• Calculate \((2)^2\). Here, the rule of exponents tells us that \(2\) raised to the power of \(2\) is \(4\).
• Next, compute \(4 \times 3\). This multiplication yields \(12\).
• Add these results together: \(4 + 12 + 1\).

The result is a simplified numerical expression, which is much easier to calculate in the final steps.

Order of operations (PEMDAS)

When simplifying an algebraic expression, it is crucial to follow the correct order of operations, often remembered by the acronym PEMDAS:

• Parentheses - Solve expressions within parentheses first.
• Exponents - Calculate powers or roots next.
• Multiplication and Division - Progress from left to right.
• Addition and Subtraction - Complete these operations, again moving from left to right.

In our example, after substituting \(x\) and simplifying, we begin with the exponents by calculating \((2)^2\). Next, we handle the multiplication \(4 \times 3\) and finally, we carry out the addition, starting with \(4 + 12\) and then adding \(1\) to get the final solution of \(17\).

Understanding and following PEMDAS ensures accurate results and helps prevent common mistakes in algebraic expression evaluation.