Question

Ujakar and Keshab attempted to solve a quadratic equation. Ujakar made a mistake in writing down the constant term. He ended up with the roots (4, 3). Keshab made a mistake in writing down the coefficient of x. He got the roots as (3, 2). What will be the exact roots of the orginal quadratic equation?

• A. (6,1)
• B. (–3, –4)
• C. (4,3)
• D. (–4, –3)
• E. (5,–3)

Answer 1

The Correct Option is A

To determine the exact roots of the original quadratic equation, we analyze the information provided about the mistakes made by Ujakar and Keshab.

1. Ujakar made a mistake in the constant term. His roots were (4, 3). The sum and product of the roots give us equations to determine the coefficients:
   • The sum of the roots = \( 4 + 3 = 7 \). This implies the coefficient of \( x \) is \(-7\).
   • The product of the roots = \( 4 \times 3 = 12 \), indicating an incorrect constant term.
2. Keshab made a mistake in the coefficient of \( x \). His roots were (3, 2). Similarly:
   • The sum of the roots = \( 3 + 2 = 5 \), suggesting a mistaken coefficient of \( x \).
   • The product of the roots = \( 3 \times 2 = 6 \), indicating the correct constant term as 6.
3. From the above, we see:
   • Ujakar provides the correct coefficient of \( x \) because his sum matches one equation: \( -7x \).
   • Keshab provides the correct constant term: \( 6 \).
4. Thus, the correct quadratic equation is represented as: x^2 - 7x + 6 = 0.
5. Solving the equation:
   • Write the factorized form: \( (x - 6)(x - 1) = 0 \).
   • The exact roots are \( x = 6 \) and \( x = 1 \).
6. Concluding, the exact roots of the original quadratic equation are (6, 1).