`(2a-5)^(2)=(4-3a)^(2)`
What is the sum of roots of the equation above?

Answer

To solve the equation \((2a - 5)^2 = (4 - 3a)^2\) and find the sum of its roots, we can follow these steps:

### Step 1: Expand both sides of the equation
We start by expanding both sides of the equation.

\[ (2a - 5)^2 = 4a^2 - 20a + 25 \]
\[ (4 - 3a)^2 = 9a^2 - 24a + 16 \]

### Step 2: Set the equation to zero
Now, we equate the two expansions:

\[ 4a^2 - 20a + 25 = 9a^2 - 24a + 16 \]

Next, we move all terms to one side to set the equation to zero:

\[ 4a^2 - 20a + 25 - 9a^2 + 24a - 16 = 0 \]

### Step 3: Combine like terms
Combining like terms gives us:

\[ (4a^2 - 9a^2) + (-20a + 24a) + (25 - 16) = 0 \]
\[ -5a^2 + 4a + 9 = 0 \]

To make it easier to work with, we can multiply through by -1:

\[ 5a^2 - 4a - 9 = 0 \]

### Step 4: Use the sum of roots formula
For a quadratic equation in the form \(ax^2 + bx + c = 0\), the sum of the roots is given by the formula:

\[ \text{Sum of roots} = -\frac{b}{a} \]

Here, \(a = 5\) and \(b = -4\). Thus, we can find the sum of the roots:

\[ \text{Sum of roots} = -\frac{-4}{5} = \frac{4}{5} \]

### Final Answer
The sum of the roots of the equation \((2a - 5)^2 = (4 - 3a)^2\) is \(\frac{4}{5}\).

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