Question

Evaluate the expression. $$3 x^{2}-5 x \text { when } x=-2$$

Short answer

The evaluated expression equals 22.

Step-by-step solution

Substitute x with -2

First, substitute every occurrence of 'x' in the expression with the given value, which is -2. The expression then becomes: \(3(-2)^{2} - 5(-2)\).

Apply Exponent

The next step is to execute the exponent in the equation. Following the PEMDAS/BODMAS rule (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), we must operate the exponent before multiplication and subtraction. Thus, \( (-2)^{2} \) is 4. Substituting in the equation results in: \(3*4 - 5*(-2)\).

Perform Multiplication

Continuing to respect the PEMDAS/BODMAS rule, we perform the multiplication operations next. This gives us: \(3*4 = 12, \) and \(-5 * -2 = 10 \). After performing these operations, the equation simplifies to \(12 + 10\).

Final Addition

Finally, add the two parts of the equation, 12 + 10. This simplifies to 22.

Key concepts

PEMDAS/BODMAS Rules

Understanding the PEMDAS/BODMAS rules is crucial for correctly evaluating algebraic expressions. These rules set the order of operations, helping you determine which calculations to perform first to get the correct result. The acronym PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. In some countries, it's known as BODMAS, which stands for Brackets, Orders (another term for exponents), Division, Multiplication, Addition, and Subtraction.

When confronted with an algebraic expression, always resolve calculations in parentheses or brackets first; this is followed by exponents or orders. Once this is done, look for multiplication or division operations, which are to be handled from left to right. The same left-to-right rule applies to addition and subtraction which are the final steps. It's essential to follow these rules step by step to avoid miscalculations and errors in algebra.

Exponentiation in Algebra

Understanding Exponents

Exponentiation is a fundamental concept in algebra where a number, known as the base, is multiplied by itself a certain number of times indicated by the exponent. For example, in the expression \(3^{2}\), 3 is the base and 2 is the exponent, meaning that 3 is multiplied by itself once: \(3 \times 3\) which equals 9.

When evaluating expressions with exponents, always remember that an exponent implies repeated multiplication. Negative bases can be tricky; if the exponent is even, the result is positive, but if the exponent is odd, the result is negative. A very important point is that exponentiation comes before multiplication and division in the order of operations. Mastering exponentiation allows you to simplify algebraic expressions effectively, ensuring that subsequent multiplication, division, addition, or subtraction is based on accurate values.

Substitution in Algebra

How Substitution Works

Substitution is a technique where you replace a variable with its corresponding value. It's particularly useful when you are given a specific value to plug into an algebraic expression or an equation. For example, if an expression includes the variable \(x\), and you are told that \(x = -2\), you would replace every \(x\) in the expression with -2 before proceeding with other operations.

Carrying out the substitution step accurately is critical because it sets the foundation for the entire evaluation process. If you substitute incorrectly, every step that follows will be impacted, leading to the wrong answer. Always double-check that you've substituted all instances of the variable and that you've correctly considered any negative signs, which could affect the outcome of exponentiation and other operations. Substitution is not just replacing numbers; it is about ensuring that these numbers interact properly according to algebraic rules within the given expression.