What is the answer to (a(3a+2)^{1/2}+2(3a+2)^3)/2 =0 ?

The answer to (a(3a+2)^{1/2}+2(3a+2)^3)/2 =0 is a\approx-0.47854…,a\approx-0.66666…

Solve for a: (a(3a+2)^{1/2}+2(3a+2)^3)/2 =0

The solution is a\approx-0.47854…,a\approx-0.66666…

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\frac{a ( 3 a + 2 ) ^{\frac{1}{2}} + 2 ( 3 a + 2 ) ^{3}}{2} = 0

a \approx - 0.47854 \dots, a \approx - 0.66666 \dots

Solution steps

\frac{a ( 3 a + 2 ) ^{\frac{1}{2}} + 2 ( 3 a + 2 ) ^{3}}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

a (3 a + 2)^{\frac{1}{2}} + 2 (3 a + 2)^{3} = 0

Expand

a (3 a + 2)^{\frac{1}{2}} + 2 (3 a + 2)^{3} : (3 a + 2)^{\frac{1}{2}} a + 54 a^{3} + 108 a^{2} + 72 a + 16

(3 a + 2)^{\frac{1}{2}} a + 54 a^{3} + 108 a^{2} + 72 a + 16 = 0

Remove square roots

3 a^{3} + 2 a^{2} = 2916 a^{6} + 11664 a^{5} + 19440 a^{4} + 17280 a^{3} + 8640 a^{2} + 2304 a + 256

Solve

3 a^{3} + 2 a^{2} = 2916 a^{6} + 11664 a^{5} + 19440 a^{4} + 17280 a^{3} + 8640 a^{2} + 2304 a + 256 : a \approx - 0.47854 \dots, a \approx - 0.66666 \dots

a \approx - 0.47854 \dots, a \approx - 0.66666 \dots

Verify Solutions:

a \approx - 0.47854 \dots

True

, a \approx - 0.66666 \dots

True

The solutions are

a \approx - 0.47854 \dots, a \approx - 0.66666 \dots

a = x + 3 + 3 x + 1

2 x + 3 - x + 2 = 2

x = (12 + 4 x)^{\frac{1}{2}}

a, a + b = 2009

What is the answer to (a(3a+2)^{1/2}+2(3a+2)^3)/2 =0 ?
The answer to (a(3a+2)^{1/2}+2(3a+2)^3)/2 =0 is a\approx-0.47854…,a\approx-0.66666…
Solve for a: (a(3a+2)^{1/2}+2(3a+2)^3)/2 =0
The solution is a\approx-0.47854…,a\approx-0.66666…