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Question

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.


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Solution

Step 1. Determine the required equations.

Assume that the numbers are xandy.

The equation to represent the difference of their squares is:

x2-y2=180

The equation to represent the second relation is:

y2=8x

Step 2. Substitute and solve.

Substitute y2=8x into x2-y2=180 and solve for y:

x2-8x=180x2-8x-180=0x-18x+10=0

x=18orx=-10

Assume that x=-10.

Substitute x=-10 into y2=8x and solve:

y2=8×-10y2=-80

The Square of a number cannot be negative, so x=18.

Substitute x=18 into y2=8x and solve:

y2=8×18y2=144y=±12

So, the larger number is 18 and the smaller number is -12or12.


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