(Q) The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

Ans: 

Step 1: Make equations:

Let the smaller number be x and larger number be y.

By 1st condition, y² – x² = 180 …. (i)

By 2nd condition, x² = 8 y… (ii)

Step 2: Solving for y:

By substituting value of x² from equation (ii) in equation (i), we get:

y² – 8 y = 180

∴ y² – 8 y – 180 = 0

∴ y² – 18 y + 10 y – 180 = 0    (by mid term splitting)

∴ y (y – 18) + 10 (y – 18) = 0

∴ (y – 18) (y + 10) = 0

∴ y = 18 and y = – 10

Step 3: Selecting value of y:

Here, we reject y = – 10 (because at y = -10, and value of x² is – 80 and we can’t take square root of negative number, we accept y = 18.

Step 4: Solving for x:

In equation (ii), we substitute y = 18, we get: 

∵ x² = 8 y

∴ x² = 8 (18)

∴ x² = 144

∴ x = 12

Therefore, smaller number be 12 and larger number be 18.

Check your answer:

If our numbers are 12 and 18,  then by 1st condition, (18)² – (12)² = 324 – 144 = 180 i.e. condition matched

(we derived value of x by 2nd condition, hence no need to check that)

Since it meets the given conditions, hence our answer is correct. 

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