Understanding the Quadratic Equation Problem

This problem involves finding the correct roots of a quadratic equation given information about incorrect roots obtained due to specific errors made by two different people, Aman and Alok. A quadratic equation is generally represented in the form \(ax^2 + bx + c = 0\), where \(a, b, c\) are coefficients and \(a \neq 0\).

Key Properties of Quadratic Equation Roots

For a quadratic equation \(ax^2 + bx + c = 0\), with roots \(\alpha\) and \(\beta\), the following properties hold:

• Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
• Product of roots: \(\alpha \times \beta = \frac{c}{a}\)

These properties are crucial for solving problems like this where information about the coefficients or roots is incomplete or incorrect.

Analyzing Aman's Mistake and Information

Aman made a mistake in writing down the constant term (\(c\)). This means the coefficient of \(x^2\) (\(a\)) and the coefficient of \(x\) (\(b\)) were written correctly by Aman. The roots Aman obtained were 4 and 3.

Since the coefficients \(a\) and \(b\) were correct, the sum of the roots calculated by Aman must be the correct sum for the original equation.

Aman's sum of roots = \(4 + 3 = 7\)

Therefore, the correct sum of the roots of the original quadratic equation is 7. Using the property of sum of roots:

\(-\frac{b}{a} = 7\)

Analyzing Alok's Mistake and Information

Alok made a mistake in writing down the coefficient of \(x\) (\(b\)). This means the coefficient of \(x^2\) (\(a\)) and the constant term (\(c\)) were written correctly by Alok. The roots Alok obtained were 3 and 2.

Since the coefficients \(a\) and \(c\) were correct, the product of the roots calculated by Alok must be the correct product for the original equation.

Alok's product of roots = \(3 \times 2 = 6\)

Therefore, the correct product of the roots of the original quadratic equation is 6. Using the property of product of roots:

\(\frac{c}{a} = 6\)

Formulating the Correct Quadratic Equation

We have the correct information from Aman's attempt (correct sum of roots) and Alok's attempt (correct product of roots):

• Correct sum of roots: \(-\frac{b}{a} = 7\)
• Correct product of roots: \(\frac{c}{a} = 6\)

We can assume the leading coefficient \(a = 1\) for simplicity. If \(a=1\), then:

• \(-\frac{b}{1} = 7 \implies -b = 7 \implies b = -7\)
• \(\frac{c}{1} = 6 \implies c = 6\)

So, the correct quadratic equation is \(x^2 + (-7)x + 6 = 0\), which simplifies to \(x^2 - 7x + 6 = 0\).

Finding the Correct Roots

Now we need to find the roots of the correct quadratic equation \(x^2 - 7x + 6 = 0\). We can solve this equation by factoring, using the quadratic formula, or completing the square. Factoring is the quickest method here.

We look for two numbers that multiply to 6 and add up to -7. These numbers are -6 and -1.

So, we can factor the equation as:

\((x - 6)(x - 1) = 0\)

Setting each factor equal to zero gives us the roots:

• \(x - 6 = 0 \implies x = 6\)
• \(x - 1 = 0 \implies x = 1\)

The correct roots of the equation are 6 and 1.

Summary of Steps

1. Identify which coefficient was correct in each person's attempt based on their mistake.
2. Use the correct roots obtained by Aman to find the correct sum of the roots for the original equation.
3. Use the correct roots obtained by Alok to find the correct product of the roots for the original equation.
4. Use the correct sum and product of roots properties \((-\frac{b}{a}\) and \(\frac{c}{a}\)) to determine the coefficients (or their ratios) of the correct quadratic equation.
5. Formulate the correct quadratic equation.
6. Solve the correct quadratic equation to find the correct roots.

Final Answer

The correct roots of the quadratic equation are 6 and 1.

Revision Table: Key Concepts Review

Additional Information: Solving Quadratic Equations

There are several methods to find the roots of a quadratic equation \(ax^2 + bx + c = 0\):

• Factoring: If the quadratic expression can be factored into the product of two linear factors, say \((px + q)(rx + s)\), the roots are found by setting each factor to zero: \(px + q = 0\) and \(rx + s = 0\).

• Using the Quadratic Formula: The roots can always be found using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The term \(\Delta = b^2 - 4ac\) is called the discriminant, which determines the nature of the roots.

• Completing the Square: This method involves transforming the equation into the form \((x - h)^2 = k\), from which the roots can be easily found by taking the square root of both sides.

Understanding the relationship between the coefficients and the roots (sum and product properties) is very useful, especially in problems where information about the equation or its roots is given indirectly, as seen in this question involving mistakes.