Question : If $a^3+3 a^2+3 a=63$, then the value of $a^2+2 a$ is:
Option 1: 22
Option 2: 19
Option 3: 15
Option 4: 8
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Correct Answer: 15
Solution :
$a^3+3 a^2+3 a=63$
We can factor the left side of the equation by recognizing it as a cube of a binomial,
$(a+1)^3=a^3+3a^2+3a+1$
Comparing this with the given equation, we have:
$(a+1)^3=63+1$
Taking the cube root of both sides, we get:
$a + 1 = \sqrt[3]{64}$
Simplifying further, we have:
$a + 1 = 4$
Subtracting 1 from both sides:
$a = 4 - 1$
⇒ $a = 3$
Now we can find the value of $a^2+2a$,
$a^2+2a = 3^2+2(3)=9+6=15$
$\therefore$ The value of $a^2+2a$ is 15
Hence, the correct answer is 15.
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