Problem 48 - Algebra 1

Question

Evaluate the expression. $$-2 x^{2}+3 x-7 \text { when } x=4$$

Step-by-Step Solution

Answer

The evaluation of the given expression when \(x = 4\) is \(-27\).

Step 1: Substitute the Variable x

Replace all instances of the variable \(x\) in the expression with the provided value, which in this case is 4. The expression therefore translates to: \(-2(4)^{2} + 3(4) - 7\).

Step 2: Observe the Order of Operations

Using the rules of BIDMAS/BODMAS, the expression simplifies as follows: First deal with the Indices/Orders, yielding \(-2*16 + 3*4 - 7\). Next, Division and Multiplication are performed from left to right resulting in \(-32 + 12 - 7\). Lastly, Addition and Subtraction are done from left to right, giving an answer of \(-32 + 5\).

Step 3: Final Evaluation

Perform the final operation to get the solution. This provides the result: \(-27\).

Key Concepts

Order of Operations

Understanding the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial when you need to evaluate algebraic expressions. When you look at the expression \( -2x^2 + 3x - 7 \) and the given value of \( x = 4 \), it's essential to apply these rules to ensure accurate calculation.

First, handle any operations inside parentheses. Next, address exponents or indices as they are second in the priority of operations. Then, perform multiplication and division as they come up, moving from left to right across the expression. Finally, tackle addition and subtraction, again working from left to right. Neglecting this hierarchy can lead to incorrect answers and is a common mistake for those learning algebra. Remembering PEMDAS and applying it reliably will help prevent such errors.

Substituting Variables

When solving algebraic expressions, substituting variables with given or known values is a fundamental skill. Take the expression \( -2x^2 + 3x - 7 \) as an example; substituting \( x \) with 4 is your first step. This transformation is essential to move from an abstract algebraic form to a concrete numerical one that can be evaluated using arithmetic.

After making the substitution \( -2(4)^2 + 3(4) - 7 \) and ensuring you have replaced every instance of \( x \), it becomes much easier to work through the expression step by step. This calculated replacement paves the way for the following steps, engaging the order of operations to eventually reach the simplified numerical answer.

Solving Quadratic Equations

Although the problem at hand does not entail solving a quadratic equation, it's valuable to understand how such equations are typically handled because they involve similar algebraic expressions. A quadratic equation takes the form \( ax^2 + bx + c = 0 \), and our expression \( -2x^2 + 3x - 7 \) could be part of such an equation if set equal to zero.

To solve a quadratic equation, you can use a variety of methods, including factoring, completing the square, or applying the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Each of these methods requires a solid grasp of algebra to manipulate the expression and isolate \( x \) for the solution. Hence, while evaluating algebraic expressions is a simpler task, it builds a foundation that is essential for tackling more complex algebraic challenges like quadratic equations.