Question

A positive real number is 1 more than another. When -2 times the smaller is added to the square of the larger, the result is 33. Find the numbers.

Answer 1

Answer:

The smaller number is 4√2 and the larger number is (4√2 + 1).

Step-by-step explanation:

Let the two numbers be x and y, where y is the larger of the two numbers.

Since y is the larger number, it is one more than the smaller number. So:

$y=x+1$

When negative two times the smaller is added to the square of the larger, the result is 33. In other words:

$-2x+y^2=33$

Substitute:

$-2x+(x+1)^2=33$

Solve for x. Square:

$-2x+(x^2+2x+1)=33$

Simplify:

$x^2+1=33$

Subtract one from both sides:

$x^2=32$

And take the square root of both sides:

$x=\pm\sqrt{32}=\pm 4\sqrt{2}$

Since y is positive, we can ignore the negative case. So, the smaller number is:

$x=4\sqrt{2}\approx5.66$

And the larger number is:

$y = 4\sqrt{2} + 1 \approx6.66$