Ram and Shyam want to solve a quadratic equation. Ram made a mistake in writing down the constant term and ended up in getting roots as 3 and 4. Shyam made a mistake in writing down the coefficient of 𝑥x and got the roots as 2 and 3. Consider the leading coefficient to be 1 in all cases. The correct roots of the quadratic equation are:

The roots of a quadratic equation are given by the formula:

x = [-b ± sqrt(b² - 4ac)] / 2a

Where:

• a is the coefficient of x²
• b is the coefficient of x
• c is the constant term

Given that the leading coefficient (a) is 1 in all cases, the formula simplifies to:

x = [-b ± sqrt(b² - 4c)] / 2

From the problem, we know that Ram made a mistake in writing down the constant term (c) and got the roots as 3 and 4. The sum and product of the roots of a quadratic equation are given by -b/a and c/a respectively. Since a is 1, the sum of the roots is -b and the product of the roots is c. Therefore, for Ram's equation:

• Sum of roots = 3 + 4 = 7 = -b
• Product of roots = 3 * 4 = 12 = c

So, b = -7 and c = 12 for Ram's equation.

Similarly, Shyam made a mistake in writing down the coefficient of x (b) and got the roots as 2 and 3. Therefore, for Shyam's equation:

• Sum of roots = 2 + 3 = 5 = -b
• Product of roots = 2 * 3 = 6 = c

So, b = -5 and c = 6 for Shyam's equation.

The correct quadratic equation should have the correct coefficient of x (b) from Ram's equation and the correct constant term (c) from Shyam's equation. Therefore, the correct quadratic equation is:

x² - 7x + 6 = 0

The roots of this equation can be found by substituting b = -7 and c = 6 into the simplified quadratic formula:

x = [7 ± sqrt((-7)² - 4*6)] / 2 x = [7 ± sqrt(49 - 24)] / 2 x = [7 ± sqrt(25)] / 2 x = [7 ± 5] / 2 x = 2, 6

Therefore, the correct roots of the quadratic equation are 2 and 6.