59.  Please solve the following equation.       1/2 x ( X-5 ) - 3/5 x ( X-2) = 2.4 *1 point3845-37-43

1. Break Down the Problem

We need to solve the equation: $\frac{1}{2} x (x - 5) - \frac{3}{5} x (x - 2) = 2.4$

2. Relevant Concepts

We will distribute the terms and combine like terms to simplify the equation before solving for $x$.

3. Analysis and Detail

1. Distribute $\frac{1}{2} x (x - 5)$: $\frac{1}{2} x^2 - \frac{5}{2} x$

2. Distribute $-\frac{3}{5} x (x - 2)$: $-\frac{3}{5} x^2 + \frac{6}{5} x$

3. Combine the terms: $\left(\frac{1}{2} x^2 - \frac{3}{5} x^2\right) + \left(-\frac{5}{2} x + \frac{6}{5} x\right) = 2.4$

4. Finding a common denominator for the $x^2$ terms: $\frac{1}{2} = \frac{5}{10}, \quad -\frac{3}{5} = -\frac{6}{10}$ Therefore, combining them: $\frac{5}{10} x^2 - \frac{6}{10} x^2 = -\frac{1}{10} x^2$

5. Finding a common denominator for the $x$ terms: $-\frac{5}{2} = -\frac{25}{10}, \quad \frac{6}{5} = \frac{12}{10}$ Thus, $-\frac{25}{10} x + \frac{12}{10} x = -\frac{13}{10} x$

6. Substitute these into the equation: $-\frac{1}{10} x^2 - \frac{13}{10} x = 2.4$

7. Multiply everything by $-10$ to clear the fractions: $x^2 + 13x + 24 = 0$

4. Verify and Summarize

We can now factor the quadratic equation: $(x + 3)(x + 8) = 0$ Setting each factor to zero gives: $x + 3 = 0 \quad \Rightarrow \quad x = -3$ $x + 8 = 0 \quad \Rightarrow \quad x = -8$

Final Answer

The solutions for the equation are: $x = -3 \quad \text{or} \quad x = -8$