Question

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Write the equation for this statement.

Hint:

When a problem involves two variables and two conditions, the goal is often to use one equation to express one variable in terms of the other, and then substitute it into the second equation to get a final equation in a single variable.

Answer 1

Step 1: Understanding the Concept:
The task is to convert the two statements given in the word problem into a single algebraic equation in one variable.

Step 2: Detailed Explanation:
Let the larger number be \(x\) and the smaller number be \(y\).
Let's translate each sentence into an equation.
Sentence 1: "The difference of squares of two numbers is 180." Since x is the larger number, this translates to: \[ x^2 - y^2 = 180 \text{--- (1)} \] Sentence 2: "The square of the smaller number is 8 times the larger number." This translates to: \[ y^2 = 8x \text{--- (2)} \] The question asks to write the equation for this statement, which usually means a single equation in one variable that models the situation. We can achieve this by substituting the expression for \(y^2\) from Equation (2) into Equation (1). \[ x^2 - (8x) = 180 \] Rearranging this into the standard form of a quadratic equation (\(ax^2+bx+c=0\)): \[ x^2 - 8x - 180 = 0 \]

Step 3: Final Answer:
This single equation, \(x^2 - 8x - 180 = 0\), represents the conditions given in the statement, where \(x\) is the larger of the two numbers.