Find a quadratic equation for which the sum of the roots is 7 and the sum of the squares of the roots is 25.

Answer

To find a quadratic equation for which the sum of the roots is 7 and the sum of the squares of the roots is 25, we can follow these steps:

### Step 1: Define the roots
Let the roots of the quadratic equation be \( a \) and \( b \).

### Step 2: Use the given information
From the problem, we know:
- The sum of the roots \( a + b = 7 \)
- The sum of the squares of the roots \( a^2 + b^2 = 25 \)

### Step 3: Relate the sum of squares to the roots
We can use the identity:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting the known values:
\[ 25 = (7)^2 - 2ab \]
This simplifies to:
\[ 25 = 49 - 2ab \]

### Step 4: Solve for \( ab \)
Rearranging the equation gives:
\[ 2ab = 49 - 25 \]
\[ 2ab = 24 \]
Dividing both sides by 2:
\[ ab = 12 \]

### Step 5: Write the quadratic equation
The standard form of a quadratic equation with roots \( a \) and \( b \) is given by:
\[ x^2 - (a + b)x + ab = 0 \]
Substituting the values we found:
\[ x^2 - 7x + 12 = 0 \]

### Final Answer
The quadratic equation is:
\[ x^2 - 7x + 12 = 0 \]

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