Question
Quantitative Aptitude Question on Algebra
If (a+b3)2=52+303, where a and b are natural numbers, then a+b equals ?
8
10
9
7
8
Solution
We are given the equation (a+b3)2=52+303, where a and b are natural numbers.
Expanding the left-hand side:
(a+b3)2=a2+2ab3+3b2
This gives us two parts: - The rational part: a2+3b2. - The irrational part: 2ab3
Equating the rational parts and the irrational parts from both sides of the equation, we get:
1. a2+3b2=52, 2. 2ab=30.
From the second equation, 2ab=30, we can solve for ab:
ab=15
Now, substitute b=a15 into the first equation:
a2+3(a15)2=52
Simplifying:
a2+a2675=52
Multiply through by a2 to clear the denominator:
a4+675=52a2
Rearranging:
a4−52a2+675=0
Let x=a2, so the equation becomes:
x2−52x+675=0
Solving this quadratic equation using the quadratic formula:
x=2×152±522−4×1×675
x=252±2704−2700
x=252±4
x=252±2
Thus, x=27 or x=25. Since x=a2, we find that a2=25, so a=5.
Now substitute a=5 into the equation ab=15:
5b=15⟹b=3
Thus, a=5 and b=3, so:
a+b=5+3=8
Therefore, the correct answer is Option (1).