To find the quadratic equation with given roots, we can use the relationship between the roots and the coefficients of a quadratic equation.

Finding the Quadratic Equation

Let the roots of the quadratic equation be \(\alpha\) and \(\beta\). The standard form of a quadratic equation with roots \(\alpha\) and \(\beta\) is given by:

\(x^2 - (\alpha + \beta)x + \alpha\beta = 0\)

Where:

• \((\alpha + \beta)\) is the sum of the roots.
• \(\alpha\beta\) is the product of the roots.

Step-by-Step Solution

1. Identify the roots: The given roots are \(\alpha = 5\) and \(\beta = 2\).

2. Calculate the sum of the roots:

   Sum \(= \alpha + \beta = 5 + 2 = 7\)

3. Calculate the product of the roots:

   Product \(= \alpha \times \beta = 5 \times 2 = 10\)

4. Substitute the sum and product into the standard equation format:

   \(x^2 - (\text{Sum})x + (\text{Product}) = 0\)

   \(x^2 - (7)x + 10 = 0\)

5. Simplify the equation:

   \(x^2 - 7x + 10 = 0\)

6. Match with the options: The derived equation \(x^2 - 7x + 10 = 0\) matches option 3.

Therefore, the quadratic equation whose roots are 5 and 2 is \(x^2 - 7x + 10 = 0\).