Question

The difference of two numbers is 4. If the sum of the smaller number and the square of the larger number is 86, what is the larger number?

Answer 1

Answer:

The larger number could be 9 or -10.

Step-by-step explanation:

Hi there!

Let the larger number be equal to a.

Let the smaller number to be equal to b.

We're given that their difference is 4:

$a-b=4$

We're also given that the sum of the smaller number and the square of the larger number is 86:

$b+a^2=86$

Rearrange the first equation to isolate a:

$a=4+b$

Substitute the first equation into the second:

$b+(4+b)^2=86\\b+(16+8b+b^2)=86\\b+16+8b+b^2=86\\16+9b+b^2=86\\b^2+9b-70=0$

Factor:

$b^2+14b-5b-70=0\\b(b+14)-5(b+14)=0\\(b-5)(b+14)=0$

The zero-product property tells us that when multiple terms have a product of 0, then at least one of the terms is equal to 0:

$b-5=0\\b=5$

Or

$b+14=0\\b=-14$

Therefore, b, or the smaller number, could be either 5 or -14.

Now, use b to solve for a:

$a-5=4\\a=9$

Therefore, the larger number is 9 if the smaller number is 5.

$a+14=4\\a=-10$

Therefore, the larger number is -10 if the smaller number is -14.

I hope this helps!