QUADRATIC EQUATIONS
The two required consecutive natural numbers are 13 and 14 respectively.
Dividing throughout by 2
14 - 13
=
1
1
1
Q. The sum of the squares of two consecutive positive numbers is
365. Find the numbers.
Sol.
Let the two required consecutive positive numbers be x and x +1.
x
+
x+1
=
365
x2 +
2x2
2x2 + 2x – 364 = 0
x2 + x – 182 = 0
x2
x
(x + 14)
x + 14 = 0
x = -14
The required numbers are natural numbers
x ≠ -14
Hence, x = 13
∴ x + 1
∴
∴
∴
∴
∴
∴
∴
∴
∴
∴
As per the given condition,
∴
∴
x2
( )2
( )2
How many numbers?
What we have to find in this sum ?
Calculation
182
91
13
2
7
13
1
7 × 2 = 14
Let us do the prime factorization of 182
Find two factors of 182 in such a way that by subtracting factors we get middle number.
| |
| |
| |
3
4
6
7
3 + 1
6 + 1
Means one after the other
+
2x
+ 1
=
365
+
2x
+
1
–
365
= 0
x
x + 1
(x + 14)
– 13
(x + 14)
= 0
(x – 13)
= 0
‘-’ sign means subtracting
Since last sign is ‘-’ Give middle sign to the bigger factor & opposite sign to smaller factor.
14
13
182
+
–
+
14x
–
13x
–182
= 0
182 × 1 = 182
or x – 13 = 0
or x = 13
= 13 + 1
= 14
EX 4.2 4
homework