Problem 44 - Fundamentals of Mathematics

Question

Find the value of each expression. $$x^{2}+3 x-1, \text { if } x=5$$

Step-by-Step Solution

Answer

The value is 39.

Step 1: Substitute the Value

Given the expression \(x^2 + 3x - 1\), we need to find its value when \(x = 5\). Start by substituting \(5\) into the expression wherever there is an \(x\). This gives us: \((5)^2 + 3(5) - 1\).

Step 2: Evaluate the Square

Calculate \(5^2\). Since \(5^2 = 25\), replace \(5^2\) with \(25\), so the expression becomes: \(25 + 3(5) - 1\).

Step 3: Multiply

Next, multiply \(3\) by \(5\) to evaluate the middle term: \(3 \times 5 = 15\). So, update the expression to \(25 + 15 - 1\).

Step 4: Simplify the Expression

Finally, perform the addition and subtraction from left to right. First, add \(25\) and \(15\) to get \(40\), then subtract \(1\) from \(40\) to get \(39\).

Key Concepts

Substitution

Substitution is a useful technique in algebra that involves replacing a variable with a specific value. In the given expression, we are working with the variable \(x\). The problem states that \(x = 5\), so the substitution process requires us to replace each occurrence of \(x\) in the expression with \(5\). Start by rewriting the expression \(x^2 + 3x - 1\) as \((5)^2 + 3(5) - 1\). This step is essential as it transforms an abstract mathematical expression into a numerical one that can be easily calculated. Substitution simplifies the evaluation process significantly by eliminating the variables, making direct computation possible.

Polynomial Evaluation

Polynomial evaluation involves calculating the value of a polynomial expression for a given value of the variable. In our exercise, the polynomial expression is \(x^2 + 3x - 1\). After substitution, this becomes a numerical expression \((5)^2 + 3(5) - 1\).

• Calculate the squared term: \((5)^2\) translates to \(5 imes 5 = 25\).
• Multiply the coefficient by the substituted value: \(3 imes 5 = 15\).

Once you've done these calculations, you combine them together in the order they appear, giving a simplified numerical expression that can be easily computed.

Order of Operations

When evaluating expressions, it's crucial to follow the order of operations, often remembered by the acronym PEMDAS:

• **P**arentheses
• **E**xponents
• **M**ultiplication and **D**ivision (from left to right)
• **A**ddition and **S**ubtraction (from left to right)

In the given expression \((5)^2 + 3(5) - 1\), we first handle exponents: \((5)^2 = 25\). Next, perform multiplication: \(3 imes 5 = 15\). Finally, carry out addition and subtraction from left to right: add \(25\) and \(15\) to get \(40\), then subtract \(1\) to end up with \(39\).
By following this order, you ensure accurate and consistent results every time you evaluate polynomial expressions.