solve problem?

Question

The larger of two natural numbers is 3 more than the smaller. The square of the smaller exceeds twice the larger by 9. Determine the numbers algebraically.

show working please

Answer 1

Let A be the smaller
Let B be the larger

Set up two equations as follows:
"The larger of two natural numbers" --> B
"is" --> =
"3 more than" --> + 3
"the smaller" --> A
B = A + 3

"The square of the smaller" --> A²
"exceeds" --> =
"twice the larger" --> 2B
"by 9" --> + 9
A² = 2B + 9

Now substitute #1 into #2:
B = A + 3
A² = 2B + 9

A² = 2(A + 3) + 9
A² = 2A + 6 + 9
A² = 2A + 15

Put everything on the left:
A² - 2A - 15 = 0

Factor:
(A - 5)(A + 3) = 0

So A = 5 or A = -3, but the question asked for *natural* numbers, so A can't be zero or negative.

A = 5

Substituting back into the original equation:
B = A + 3
B = (5) + 3
B = 8

Double-checking:
The larger (8) is 3 more than the smaller (5).
The square of the smaller (25) is twice the larger (16) plus 9.

So your numbers are 5 and 8.

Answer 2

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

x+3=y from the first sentence

X2 (read x squared) =2y+9. solve simultaneously
2y=x2-9
y=x+3

multiply the bottom equation by 2 and then subtract from the top
2y=x2-9
-2y=2x+6
equals 0=x2-2x-15
FOIL factor

(x )(x )
(x 5)(x 3)
because the sign on the 15 is negative then the signs are different and the negative goes with the larger number because the sign on the 2x is negative.
(x-5)(x+3)=0

x=5, and x=-3.

Check your answer if x=5 then y =8 because it must be 3 more larger than the smaller. 2(8)+9 = 25 and 5 squared = 25 so yep 5 works.

Check your definition for natural numbers. We were always taught they did not include negatives. If that is the case then the -3 is automatically disqualified. Otherwise check this answer on your on.

Answer 3

let the numbers be x and (x+3)

so, x^2-2(x+3)=9
=> x^2-2x-15=0
=> (x-5)(x+3)=0
=> x=5 or -3
x being natural, we ignore x=-3
so, the reqd numbers are 5 and 8

Answer 4

L=S+3
S^2=L+9
(L-3)^2=2L+9
L^2-6L+9=2L+9
L^2-8L=0
L(L-8)=0
L=8
S=5
the nos are 5 and 8