By what amount does the sum of the roots exceed the product of the roots of the equation `(x-5)(x+2)=0`?

Answer

To solve the question, we need to find the roots of the equation `(x - 5)(x + 2) = 0`, calculate the sum and product of the roots, and then determine by what amount the sum exceeds the product.

### Step 1: Find the roots of the equation
The given equation is:
\[ (x - 5)(x + 2) = 0 \]
To find the roots, we set each factor equal to zero:
1. \(x - 5 = 0 \Rightarrow x = 5\)
2. \(x + 2 = 0 \Rightarrow x = -2\)

So, the roots of the equation are \(5\) and \(-2\).

### Step 2: Calculate the sum of the roots
The sum of the roots is calculated as follows:
\[ \text{Sum of roots} = 5 + (-2) = 5 - 2 = 3 \]

### Step 3: Calculate the product of the roots
The product of the roots is calculated as:
\[ \text{Product of roots} = 5 \times (-2) = -10 \]

### Step 4: Determine by what amount the sum exceeds the product
We need to find the difference between the sum and the product:
\[ \text{Difference} = \text{Sum of roots} - \text{Product of roots} = 3 - (-10) \]
This simplifies to:
\[ 3 + 10 = 13 \]

### Final Answer
The sum of the roots exceeds the product of the roots by \(13\).

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