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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