What is the value of a?



a–2=3+6a3



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1 answer

To solve the equation \( a - 2 = 3 + 6a^3 \), we will rearrange it to find the value of \( a \).

Starting with the original equation:

\[ a - 2 = 3 + 6a^3 \]

First, we can move all terms to one side of the equation:

\[ a - 2 - 3 - 6a^3 = 0 \]

This simplifies to:

\[ a - 5 - 6a^3 = 0 \]

Rearranging gives:

\[ 6a^3 + a - 5 = 0 \]

Now we can apply numerical methods or graphing methods to find the roots. Alternatively, we can also check for rational roots using the Rational Root Theorem. Possible rational roots are factors of \(-5\) (i.e., \(\pm 1, \pm 5\)).

Let's check \( a = 1 \):

\[ 6(1^3) + 1 - 5 = 6 + 1 - 5 = 2 \quad (\text{not a root}) \]

Next, check \( a = -1 \):

\[ 6(-1^3) + (-1) - 5 = -6 - 1 - 5 = -12 \quad (\text{not a root}) \]

Next, check \( a = 5 \):

\[ 6(5^3) + 5 - 5 = 6(125) + 5 - 5 = 750 + 5 - 5 = 750 \quad (\text{not a root}) \]

Finally, check \( a = -5 \):

\[ 6(-5^3) + (-5) - 5 = 6(-125) - 5 - 5 = -750 - 5 - 5 = -760 \quad (\text{not a root}) \]

Since these rational guesses did not yield roots, we could either use numerical methods or graphing to approximate.

Let's examine the behavior of the function \( f(a) = 6a^3 + a - 5 \):

Using numerical approximation techniques or solving via a calculator, we can find:

The real root of the equation \( 6a^3 + a - 5 = 0 \) is approximately \( a \approx 0.710 \).

Therefore, the approximate value of \( a \) is:

\[ \boxed{0.710} \]