Write the discriminant of the quadratic equation `(x+5)^(2)=2(5x-3)`.

Answer

To find the discriminant of the quadratic equation \((x+5)^{2}=2(5x-3)\), we will follow these steps:

### Step 1: Expand the left-hand side
We start with the equation:
\[ (x + 5)^2 = 2(5x - 3) \]
Expanding the left-hand side using the formula \((a + b)^2 = a^2 + 2ab + b^2\):
\[ x^2 + 10x + 25 = 2(5x - 3) \]

### Step 2: Expand the right-hand side
Now we expand the right-hand side:
\[ 2(5x - 3) = 10x - 6 \]
So the equation now looks like:
\[ x^2 + 10x + 25 = 10x - 6 \]

### Step 3: Rearrange the equation
Next, we will move all terms to one side of the equation:
\[ x^2 + 10x + 25 - 10x + 6 = 0 \]
This simplifies to:
\[ x^2 + 31 = 0 \]

### Step 4: Identify coefficients
Now, we can identify the coefficients \(a\), \(b\), and \(c\) from the standard form of a quadratic equation \(Ax^2 + Bx + C = 0\):
- \(a = 1\)
- \(b = 0\)
- \(c = 31\)

### Step 5: Calculate the discriminant
The discriminant \(D\) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values we found:
\[ D = 0^2 - 4 \cdot 1 \cdot 31 \]
Calculating this gives:
\[ D = 0 - 124 = -124 \]

### Final Answer
The discriminant of the quadratic equation \((x+5)^{2}=2(5x-3)\) is:
\[ D = -124 \]

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