One number is equal to the square of another....

Question

One number is equal to the square of another. Find the numbers if their sum is 72.

Answer 1

When a problem uses terms like "a number" or "some value" it is best to replace them with variables to set up equations in order to solve.

"One number is equal to the square of another." can be rewritten with variables as x=y^2

Find the numbers if their sum is 72 means that x+y=72

Now we have a system of equations:

x = y^2

x + y = 72

Let's solve by substitution for x since the first equation is already solved for x

Plugging in to the second equation:

y^2 + y = 72

Now, let's factor:

y^2 + y = 72

-72 -72

y^2 + y - 72 = 0

(y - 8)(y + 9) = 0

y = -9, 8

plugging y back into the second equation to find x

x + -9 = 72 x + 8 = 72

x = 81 x = 64

Since -9^2 = 81 and -9 + 81 = 72, a solution (81, -9) exists

Since 8^2 = 64 and 8 + 64 = 72, a solution (64, 8) exists as well.

Answer 2

Let x and y be the two numbers.

1. One number is equal to the square of the other: x = y²
2. Their sum is 72: x + y = 72

So you have two unknowns (x and y) and two equations.

x = y²

x + y = 72

Since x = y², let's substitute y² in place of x in the second equation:

x + y = 72

y² + y = 72

y² + y - 72 = 0

You can solve this quadratic by factoring (or by using the quadratic formula). You'll get two answers, both of which are correct. Once you have the values of y, find the values of x using x = y².