Question

$$\text { If } f(x)=3 x^{2}-25 x+28, \text { find } f(7)$$

Short answer

f(7) = 0

Step-by-step solution

Identify the function

The given function is defined as \( f(x) = 3x^2 - 25x + 28 \).

Substitute the value of x

To find \( f(7) \), substitute \( 7 \) for \( x \) in the function: \( f(7) = 3(7)^2 - 25(7) + 28 \).

Calculate \( 3(7)^2 \)

First, calculate the term \( 3(7)^2 \): \( 3(7)^2 = 3 \times 49 = 147 \).

Calculate \( -25(7) \)

Next, calculate the term \( -25(7) \): \( -25(7) = -175 \).

Sum the results

Finally, add all the terms together: \( 147 - 175 + 28 \).

Compute the final value

Complete the addition and subtraction: \( 147 - 175 + 28 = 0 \).

Key concepts

Quadratic Function

A quadratic function is a type of polynomial function that has the form: ewline

Here, a, b, and c represent constants, and x is the variable. The highest degree of the variable x in a quadratic function is 2, giving it the name 'quadratic'. ewlineIn the given problem, the quadratic function is: ewline

Notice that it fits the general form, where a = 3, b = -25, and c = 28.Quadratic functions are frequently used in various fields like physics, engineering, and economics to describe phenomena involving parabolic relationships. Understanding the basic structure and behavior of these functions is crucial for solving related problems effectively.

Function Evaluation

Function evaluation is the process of finding the value of a function for a specific input, usually given as f(x) for some x. This involves substituting the given value of x into the function and simplifying the expression. ewlineIn our problem, we are asked to find the value of f(7) for the function: ewline

To do this, we substitute x with 7: ewline

This step-by-step substitution is crucial for accurate results. After substituting, we follow through with the calculations, which involve multiple arithmetic operations.

Substitution Method

The substitution method is a fundamental technique widely used in algebra, calculus, and other areas of mathematics. It helps in solving equations and evaluating expressions by replacing a variable with a given number. ewline

• Start with the given function or equation.
• Replace the variable with the provided value.
• Simplify the resulting expression step-by-step to find the solution.

In the original problem, we use the substitution method to find f(7): ewlineFirst, replace x with 7 in the function: ewlineNext, calculate each term individually: ewline

• 3(7)^2 = 3 * 49 = 147
• -25(7) = -175

Finally, add all the terms together: ewline

• 147 - 175 + 28 = 0

Using the substitution method correctly ensures that we get an accurate and simplified result for complex expressions.