a sum of a positive integer and its square is 90 . find the number
Answers
Answer:
9
Explanation:
Let
n
be the integer in question. Then we have
n
2
+
n
=
90
⇒
n
2
+
n
−
90
=
0
We now have a quadratic equation to solve. We could use the quadratic formula, however we know that
n
is an integer, so instead let's try to solve by factoring instead.
n
2
+
n
−
90
=
0
⇒
n
2
+
10
n
−
9
n
−
90
=
0
⇒
n
(
n
+
10
)
−
9
(
n
+
10
)
=
0
⇒
(
n
−
9
)
(
n
+
10
)
=
0
⇒
n
−
9
=
0
or
n
+
10
=
0
⇒
n
=
9
or
n
=
−
10
As it is given that
n
>
0
, we can disregard the possibility that
n
=
−
10
, leaving us with our final answer of
n
=
9
Checking our result, we find that it satisfies the given conditions:
9
+
9
2
=
9
+
81
=
90
Let the positive integer be x
The sum of the positive integer and its square is 90:
x + x² = 90
x² + x - 90 = 0
Solve the quadratic equation by factoring:
(x + 10) (x - 9) = 0
x + 10 = 0 x - 9 = 0
x = -10 x = 9
Choose the positive root, x= 9.
The positive integer is 9.
To check:
x + x² = 90
9 + (9)² = 90
9 + 81 = 90
90 = 90