Solving the Algebraic Equation

We are asked to find the value of $x$ that satisfies the equation: $3(2x^2 - 5) - 2(3x^2 + 4x - 8) = 20$

Step-by-Step Solution

1. Expand the terms:

   First, distribute the constants outside the parentheses into the terms inside them.

   • Multiply 3 by each term in the first parenthesis: $3 \times 2x^2 = 6x^2$ and $3 \times (-5) = -15$.
   • Multiply -2 by each term in the second parenthesis: $-2 \times 3x^2 = -6x^2$, $-2 \times 4x = -8x$, and $-2 \times (-8) = 16$.

   Substituting these back into the equation gives:

   $6x^2 - 15 - 6x^2 - 8x + 16 = 20$
2. Simplify the equation:

   Combine like terms on the left side of the equation.

   • Combine the $x^2$ terms: $6x^2 - 6x^2 = 0$. These terms cancel each other out.
   • Combine the constant terms: $-15 + 16 = 1$.
   • The term with $x$ is $-8x$.

   After combining like terms, the equation simplifies to:

   $-8x + 1 = 20$
3. Isolate the x term:

   To isolate the term containing $x$, subtract 1 from both sides of the equation.

   $-8x + 1 - 1 = 20 - 1$ $-8x = 19$
4. Solve for x:

   Finally, divide both sides by -8 to find the value of $x$.

   $\frac{-8x}{-8} = \frac{19}{-8}$ $x = -\frac{19}{8}$

Final Answer Verification

Let's check if $x = -\frac{19}{8}$ satisfies the original equation:

The value $x = -\frac{19}{8}$ satisfies the equation.

Conclusion

The value of $x$ that satisfies the given equation $3(2x^2 - 5) - 2(3x^2 + 4x - 8) = 20$ is $x = -\frac{19}{8}$.