Quadratic Equation

Step 1: Define the integers.

Let the smaller integer be . Then the next consecutive integer is .

Step 2: Write the product condition.

The product of the two integers is:

Expand:

Rearrange into standard quadratic form:

Final Answer: The quadratic representation is , which corresponds to option .

Step 1: Define variables.

Let the time taken by B to finish the work be days. Then the time taken by A is days.

Step 2: Express their work rates.

The work rate of B is (work per day), and the work rate of A is .

Together, their combined work rate is (since they finish the work in 4 days).

Step 3: Set up the equation.

Step 4: Solve the equation.

Multiply through by to eliminate the denominators:

Simplify:

Factorize:

Thus, or . Since would make negative (impossible), we have .

Final Answer: The time taken by B to finish the work is , which corresponds to option .

Step 1: Recall the condition for equal roots.

A quadratic equation has equal roots if its discriminant is zero.

Step 2: Compute the discriminant.

Here, , , and . The discriminant is:

Simplify:

Step 3: Set the discriminant to zero and solve for .

Final Answer: The value of is , which corresponds to option .