Question

If \(x=-2,\) the value of the expression \(3 x^{2}-5 x+\frac{1}{x}\) is ___________

Short answer

The value of the expression is 21.5.

Step-by-step solution

Substitute the value of x

Given that \(x=-2\), substitute \(-2\) wherever \(x\) appears in the expression. The expression becomes: \[3(-2)^{2}-5(-2)+\frac{1}{-2}\]

Evaluate the exponent

Calculate \((-2)^{2}\). Remember that raising a negative number to an even exponent results in a positive number. \((-2)^{2} = 4\), so the expression is now: \[3 \cdot 4 - 5 (-2) + \frac{1}{-2}\]

Simplify multiplications

Multiply \(3\) by \(4\): \[3 \cdot 4 = 12\]. Multiply \(-5\) by \(-2\): \[-5 \cdot -2 = 10\]. The expression simplifies to: \[12 + 10 + \frac{1}{-2}\]

Simplify the fraction

Calculate the fraction \(\frac{1}{-2}\), which is \(-\frac{1}{2}\). The expression is now: \[12 + 10 - \frac{1}{2}\]

Sum the values

Combine the simplified terms: \[12 + 10 = 22\]. Subtract \(\frac{1}{2}\) from \(22\): \[22 - \frac{1}{2} = 21.5\]

Key concepts

Substitution Method

The substitution method involves replacing a variable with a given value. In the exercise, we are given that \( x = -2 \). To find the value of the expression \( 3x^2 - 5x + \frac{1}{x} \), we need to substitute every \( x \) in the expression with \( -2 \). This transforms the original expression to \( 3(-2)^2 - 5(-2) + \frac{1}{-2} \). Substitution is crucial in solving algebra problems, especially when you have specific values for the variables.

Simplifying Expressions

Simplifying expressions helps make complex problems more manageable. When we simplified the given expression, we had to:

• Evaluate exponents
• Multiply terms
• Combine like terms

For instance, in our exercise, after substitution, we simplified \( 3(-2)^2 \) to \( 3 \cdot 4 \) which equals 12. Similarly, we simplified the multiplication \( -5(-2) \) to 10. Simplifying helps in breaking down the problem into smaller, more solvable parts.

Evaluating Exponents

Exponents indicate how many times a number is multiplied by itself. In the expression \( 3(-2)^2 \), \(-2\) is raised to the power of 2. It is key to remember that raising a negative number to an even power results in a positive number: \( (-2)^2 = 4 \). The expression thus changes to \( 3 \cdot 4 \), simplifying it further. Mastering exponents is essential for understanding and solving higher-level algebra problems.

Fraction Operations

Working with fractions requires careful attention, especially when involving negative signs. In our problem, after simplifying the other parts of the expression, we were left with the term \( \frac{1}{-2} \). This simplifies to \( -\frac{1}{2} \). When combining this with other parts of the expression, it's important to handle the sign correctly. Finally, adding \( 22 - \frac{1}{2} \) gives us \( 21.5 \). Understanding how to handle and simplify fractions is key in algebraic operations.