Question

The product of Karan’s age five years ago and his age after 9 years from now is 32. This is represented by the quadratic equation:

• A. \( x^2 + 4x + 77 = 0 \)
• B. \( x^2 - 4x + 77 = 0 \)
• C. \( x^2 + 4x - 77 = 0 \)
• D. \( x^2 - 4x - 77 = 0 \)

Hint:

To set up quadratic equations from word problems involving age, translate the relationships into expressions for past and future ages, then multiply them to form the equation.

Answer 1

The Correct Option is D

Step 1: Let Karan’s present age be \( x \)
The problem states that the product of Karan’s age five years ago and his age after nine years is 32.
Karan’s age five years ago is \( x - 5 \).
Karan’s age after nine years from now is \( x + 9 \).
Step 2: Set up the equation
According to the problem: \[ (x - 5)(x + 9) = 32 \] Step 3: Expand the equation
Expanding the left-hand side: \[ x^2 + 9x - 5x - 45 = 32 \] \[ x^2 + 4x - 45 = 32 \] Step 4: Simplify the equation
Subtract 32 from both sides: \[ x^2 + 4x - 45 - 32 = 0 \] \[ x^2 + 4x - 77 = 0 \] Step 5: Conclusion
Thus, the quadratic equation representing the problem is \( x^2 + 4x - 77 = 0 \), which corresponds to option \( (4) \).