Writing a Quadratic Equation to Represent a Problem
When the product of 6 and the square of a number is increased by 5 times the number, the result is 4. Which equation represents this situation?
$6 x^2+5 x+4=0$
$6^2 x+5+x=4$
$6 x^2+5 x-4=0$
$6+x^2+5 x=4
Answer (1)
Represent the unknown number with the variable x .
Translate 'the product of 6 and the square of a number' to 6 x 2 .
Translate 'increased by 5 times the number' to + 5 x .
Set the expression equal to 4 and rearrange to get the quadratic equation: 6 x 2 + 5 x − 4 = 0 .
Explanation
Understanding the Problem Let's break down the problem step by step to form the correct equation.
Defining the Variable We are given that 'the product of 6 and the square of a number' is increased by '5 times the number', and the result is 4. Let's use 'x' to represent the number.
Expressing the First Part The product of 6 and the square of the number can be written as 6 x 2 .
Expressing the Second Part '5 times the number' can be written as 5 x .
Combining the Expressions The problem states that 6 x 2 is increased by 5 x , which means we add them together: 6 x 2 + 5 x .
Forming the Equation Finally, we are told that this sum equals 4, so we have the equation 6 x 2 + 5 x = 4 . To write this in the standard quadratic form, we subtract 4 from both sides to get 6 x 2 + 5 x − 4 = 0 .
Final Equation Therefore, the equation that represents the given situation is 6 x 2 + 5 x − 4 = 0 .
Examples
Imagine you're designing a rectangular garden where the area is determined by the equation 6 x 2 + 5 x − 4 = 0 . Here, 'x' could represent a design parameter, and solving the equation helps you find the specific dimensions that satisfy your area requirements. Quadratic equations like this are fundamental in various fields, including physics, engineering, and economics, for modeling and solving real-world problems involving optimization and relationships between variables.
