Question

The difference of squares of two numbers is 180. If the square of the smaller number is 8 times the larger number, find both numbers.

Hint:

Word problems → form equations first. Steps:
Assign variables clearly.
Convert statements into equations.
Substitute to reduce variables.
Always check which root is valid.

Answer 1

Step 1: Understanding the Question:
This is a word problem describing a relationship between two unknown numbers. Our task is to translate the given information into a system of algebraic equations and solve them to find the numbers.
Step 2: Key Formula or Approach:
1. Define variables to represent the unknown numbers (e.g., \(x\) for the smaller number and \(y\) for the larger number).
2. Formulate two equations based on the two conditions given in the problem statement.
3. Solve the system of equations, likely using the substitution method, which will lead to a quadratic equation.
4. Solve the quadratic equation and check the validity of the solutions in the context of the problem.
Step 3: Detailed Explanation:
Let the larger number be \(y\) and the smaller number be \(x\).
Condition 1: "The difference of squares of two numbers is 180."
This can be written as the equation:
\[ y^2 - x^2 = 180 \quad \cdots (1) \] Condition 2: "The square of the smaller number is 8 times the larger number."
This can be written as the equation:
\[ x^2 = 8y \quad \cdots (2) \] Now we have a system of two equations. We can use substitution to solve it. Substitute the expression for \(x^2\) from equation (2) into equation (1):
\[ y^2 - (8y) = 180 \] Rearrange this into the standard quadratic form \(ay^2 + by + c = 0\):
\[ y^2 - 8y - 180 = 0 \] We can solve this quadratic equation by factoring. We are looking for two numbers that multiply to -180 and add to -8. These numbers are -18 and +10.
\[ (y - 18)(y + 10) = 0 \] This gives two possible solutions for \(y\): \(y = 18\) or \(y = -10\).
Let's check each case:
Case A: If $y = -10$.
Substitute this into equation (2): \(x^2 = 8(-10) = -80\). The square of a real number cannot be negative, so this solution is not valid.
Case B: If $y = 18$.
Substitute this into equation (2): \(x^2 = 8(18) = 144\).
Now, we find \(x\) by taking the square root:
\[ x = \sqrt{144} \implies x = \pm 12 \] Since the problem refers to "smaller" and "larger" numbers without specifying their sign, both \(x=12\) and \(x=-12\) are mathematically valid. Typically in such problems, positive numbers are intended. Let's take the positive value \(x=12\).
Step 4: Final Answer:
The larger number is 18 and the smaller number is 12.
\[ \boxed{\text{Larger number } = 18, \quad \text{ Smaller number } = 12} \]