Problem 7 - Algebra and Trigonometry

Question

An expression is given. (a) Evaluate it at the given value. (b) Find its domain. $$ 4 x^{2}-10 x+3, \quad x=5 $$

Step-by-Step Solution

Answer

(a) The expression evaluates to 53 at \(x = 5\). (b) The domain is all real numbers, \(\mathbb{R}\).

Step 1: Substitute the Given Value

Substitute \(x = 5\) into the expression \(4x^2 - 10x + 3\). This results in \(4(5)^2 - 10(5) + 3\).

Step 2: Evaluate the Expression

Calculate \(4(5)^2 = 4 \times 25 = 100\). Subtract \(10 \times 5 = 50\) and add 3: thus the expression becomes \(100 - 50 + 3\).

Step 3: Simplify the Result

Simplify the expression: \(100 - 50 + 3 = 53\). Therefore, the value of the expression at \(x = 5\) is 53.

Step 4: Identify the Domain of the Expression

The expression is a polynomial (quadratic in this case), which is defined for all real numbers. Therefore, the domain of the expression is all real numbers \(\mathbb{R}\).

Key Concepts

Quadratic Expression

A quadratic expression is a type of polynomial and is characterized by the highest power of the variable being 2. In simple terms, it looks like \[ ax^2 + bx + c \] where \( a \), \( b \), and \( c \) represent constants, with \( a eq 0 \). Quadratic expressions are common in algebra and can be visualized as a parabola on a graph.

• The leading coefficient is \( a \).
• The term \( bx \) is the linear component.
• The constant \( c \) does not involve any variables.

These expressions appear in various scenarios, like calculating areas, projectile motion, and solving real-world problems. Recognizing and manipulating them is fundamental in mathematics.
Understanding the form and components of a quadratic expression is crucial when evaluating these expressions for specific values.

Substitution

Substitution is a method used in algebra to evaluate expressions for particular values. It involves replacing the variable in an expression with a given number. For example, in the expression \( 4x^2 - 10x + 3 \), if we need to evaluate it at \( x = 5 \), we "substitute" 5 in place of \( x \).
Here's how it's done:

• Replace each \( x \) in the expression with the given number, for instance, 5. This results in \( 4(5)^2 - 10(5) + 3 \).
• Perform the calculations following the correct order of operations (first, handle the exponent, then multiplication, followed by addition and subtraction).

Substitution allows us to find the value of the expression for specific values of the variable, providing insights into how the expression behaves under different circumstances.

Polynomial Domain

The domain of a polynomial is the set of all possible values that the variable(s) can take. For most polynomial expressions, the domain is quite straightforward because they are defined for all real numbers.
In mathematical terms:

• Polynomials in one variable, like the quadratic \( 4x^2 - 10x + 3 \), have a domain of all real numbers, denoted by \( \mathbb{R} \).
• This means any real number can be substituted into the variable "x" without causing any mathematical problems, like division by zero.

Understanding the domain is essential as it tells us the scope of inputs that can be used for evaluation without encountering undefined situations. Knowing the domain also aids in solving equations involving these expressions.

Real Numbers

Real numbers encompass a broad range of numbers you encounter daily, including all integers, fractions, and decimals, both rational and irrational numbers. They're represented by the symbol \( \mathbb{R} \). Real numbers play a crucial role in defining the domain of many mathematical expressions, including polynomials.
Key aspects of real numbers include:

• They're continuous and unbroken, allowing any subtraction, addition, multiplication, and division, except by zero.
• This set includes negative, zero, and positive values. Examples are -3, 0, 4.56, and \( \sqrt{2} \).

With real numbers, we can evaluate polynomial expressions like the quadratic form for a wide range of values, making them fundamental in both theoretical and applied mathematics. Having the domain of a polynomial as all real numbers means there are no limitations related to the types of values we can substitute in place of the variable.