Question

Evaluate the expression. $$2 x^{2}+8 x-5 \text { when } x=3$$

Short answer

The value of the expression \(2x^{2}+8x-5\) when \(x=3\) is 37

Step-by-step solution

Substitute the given value into the equation

In the given expression \(2x^{2}+8x-5\), substitute \(x\) with 3. So we get \(2(3)^{2}+8(3)-5\).

Simplify the equation

Perform the exponent operation first, according to the order of operations which encompasses BIDMAS (Brackets, Indices, Division and Multiplication, Addition and Subtraction). So we get \(2*(9)+8(3)-5\), which becomes \(18+24-5\).

Perform addition and subtraction

Perform the addition and then the subtraction: \(18+24-5 = 42-5 = 37\).

Key concepts

Substitution Method

In algebra, the substitution method is fundamental in evaluating expressions. It simply involves replacing variables with their given numerical values.

For example, if an expression includes a variable, say 'x', and you are given that 'x equals 3', you substitute every occurrence of 'x' in the expression with the number '3'. It’s important to retain the original structure of the expression while doing so. This is your first step towards finding the value of the algebraic expression.

Applying this to our exercise, we begin by taking the expression \(2x^{2}+8x-5\) and substituting '3' in place of 'x', which gives us \(2(3)^{2}+8(3)-5\). This replacement sets the stage for further simplifications using order of operations.

Order of Operations

After substituting the variables with numbers, the next step is to apply the order of operations. This is crucial because it dictates the sequence in which the operations within an expression should be performed to guarantee the correct result.

The acronym PEMDAS (or BIDMAS in some regions), which stands for Parentheses (or Brackets), Exponents (or Indices), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right), can help you remember this sequence. The order of operations is vital because different sequences can lead to different results, which could be incorrect.

For the given example \(2(3)^{2}+8(3)-5\), we first address the exponent, resulting in \(2*9\). Following this, we carry out the multiplication and then the addition and subtraction in sequence, leading to the correct result.

Simplifying Equations

The final stage in evaluating algebraic expressions is simplifying the equation. Simplification includes carrying out all the operations in the expression according to the order of operations, and reducing it to its simplest form, which is usually a single number or a simpler expression.

As part of simplification, once you complete the operations associated with exponents and perform any required multiplication or division, you finalize with addition or subtraction. Each step must be completed in isolation, ensuring that the full operation is finished before moving on to the next.

In our exercise, after applying the exponent, we had \(2*9+8*3-5\), which simplifies to \(18+24-5\). The last steps involve adding 18 and 24 to get 42, and then subtracting 5, yielding the simplified result of 37.