Problem 48 - Fundamentals of Mathematics

Question

Find the value of each expression. $$16 x^{2}+8 x-7, \text { if } x=0$$

Step-by-Step Solution

Answer

-7

Step 1: Substitute the value of x

We need to evaluate the expression for \( x = 0 \). Substitute \( x = 0 \) into the expression. This gives us:\[ 16(0)^{2} + 8(0) - 7. \]

Step 2: Calculate powers of x

Calculate the power of \( x \), which is \( 0^2 \). We have:\[ 0^2 = 0. \]

Step 3: Multiply coefficients by x terms and add

Using the result from Step 2, we multiply the terms:\[ 16(0) = 0 \ 8(0) = 0. \]

Step 4: Simplify the expression

Now that we have substituted and multiplied, add the results:\[ 0 + 0 - 7 = -7. \]

Key Concepts

Substitution Method

The substitution method is a straightforward technique used to evaluate algebraic expressions. In this process, we replace the variable with a specific number. In our original exercise, we're finding the value of the expression when \( x = 0 \). By substituting \( x \) with 0, the expression \( 16x^2 + 8x - 7 \) turns into \( 16(0)^2 + 8(0) - 7 \). This transformation allows us to focus on calculations without variables.

• It helps simplify the work by turning variables into constants.
• The method is useful in both simple and complex algebraic problems.
• It lays the foundation for more advanced mathematical concepts.

To apply this technique efficiently, always ensure you've substituted every instance of the variable in your expression.

Algebraic Expressions

Algebraic expressions, like the one in the problem \( 16x^2 + 8x - 7 \), are composed of variables, numbers, and operators. They form the basis of algebra and allow us to model real-life situations mathematically. Understanding the parts of an algebraic expression is key:

• Terms: Separate parts of the expression, like \( 16x^2 \), \( 8x \), and \(-7 \).
• Coefficients: Numbers multiplying the variables, such as 16 and 8 in our example.
• Variables: Usually represented by letters like \( x \), indicating unknown values.
• Constants: Numbers on their own, like \(-7\) in the expression.

By getting comfortable with these components, you will more easily navigate through evaluating and manipulating algebraic expressions.

Simplification Steps

Simplification is the process of reducing an expression to its simplest form. After substituting the variable in our expression and calculating the powers and products, we arrive at the simplified form \( 0 + 0 - 7 \). The steps followed were:

• Substituting the variable \( x \) with 0 to remove it from the expression.
• Finding the power of \( x \), such as \( 0^2 = 0 \).
• Multiplying coefficients by the result of the substituted variable, resulting in terms like \( 16(0) = 0 \).
• Adding up all resulting terms to find the simplest form.

An important tip is to follow the order of operations: parentheses, exponents, multiplication and division, addition and subtraction (PEMDAS). Simplification not only makes expressions easier to handle but also ensures accuracy in solving complex mathematical equations.