Understanding the Quadratic Equation Problem

This problem involves finding the correct quadratic equation when information about incorrect roots obtained due to specific mistakes in the equation is provided. A standard quadratic equation can be written in the form $ax^2 + bx + c = 0$. If we divide by $a$ (assuming $a \neq 0$), the equation becomes $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$. Let's denote $p = \frac{b}{a}$ and $q = \frac{c}{a}$, so the equation is $x^2 + px + q = 0$.

For a quadratic equation $x^2 + px + q = 0$, with roots $\alpha$ and $\beta$, the following relationships hold:

• Sum of roots: $\alpha + \beta = -p$
• Product of roots: $\alpha \beta = q$

We are given information about two students who made mistakes while solving the original quadratic equation. Each student made a mistake in only one specific part of the equation, leading to incorrect roots. We will use the roots they obtained to deduce the correct coefficients of the original quadratic equation.

Analyzing Student 1's Mistake and Roots

The first student made a mistake in the constant term ($q$) of the quadratic equation. This means that the coefficient of the first-degree term ($p$) was correct in the equation they solved.

The roots obtained by this student are 8 and 2.

Let the equation the first student solved be $x^2 + p_{correct}x + q_{incorrect} = 0$.

For this equation, the sum of the roots is $8 + 2 = 10$.

According to the sum of roots formula, the sum is equal to $-p_{correct}$.

So, $-p_{correct} = 10$.

This gives us $p_{correct} = -10$.

Since the student's mistake was only in the constant term, the coefficient of the first-degree term, $p$, from their equation is the correct $p$ for the original equation.

Therefore, the correct coefficient of the first-degree term is $-10$.

Analyzing Student 2's Mistake and Roots

The second student made a mistake only in the coefficient of the first-degree term ($p$) of the quadratic equation. This means that the constant term ($q$) was correct in the equation they solved.

The roots obtained by this student are -9 and -1.

Let the equation the second student solved be $x^2 + p_{incorrect}x + q_{correct} = 0$.

For this equation, the product of the roots is $(-9) \times (-1) = 9$.

According to the product of roots formula, the product is equal to $q_{correct}$.

So, $q_{correct} = 9$.

Since the student's mistake was only in the coefficient of the first-degree term, the constant term, $q$, from their equation is the correct $q$ for the original equation.

Therefore, the correct constant term is $9$.

Constructing the Correct Quadratic Equation

We have determined the correct coefficient of the first-degree term ($p = -10$) and the correct constant term ($q = 9$) for the quadratic equation in the form $x^2 + px + q = 0$.

Substituting these values, we get:

$x^2 + (-10)x + 9 = 0$

Simplifying this equation gives:

$x^2 - 10x + 9 = 0$

This is the correct quadratic equation that was intended to be solved.

Verification of the Correct Quadratic Equation

Let's check if the equation $x^2 - 10x + 9 = 0$ yields the roots from the incorrect solutions if specific terms are changed.

• Correct roots: To find the roots of $x^2 - 10x + 9 = 0$, we can factor it: $(x-1)(x-9)=0$. The correct roots are $x=1$ and $x=9$.
• Student 1's case (mistake in constant term): The student got roots 8 and 2. The equation based on these roots would have a sum $8+2=10$ and product $8 \times 2 = 16$. The equation would be $x^2 - 10x + 16 = 0$. Comparing this to the correct equation $x^2 - 10x + 9 = 0$, the coefficient of the first-degree term (-10) is correct, and only the constant term (16 instead of 9) is different. This matches the problem description for student 1.
• Student 2's case (mistake in coefficient of first-degree term): The student got roots -9 and -1. The equation based on these roots would have a sum $(-9)+(-1)=-10$ and product $(-9) \times (-1) = 9$. The equation would be $x^2 - (-10)x + 9 = 0$, which is $x^2 + 10x + 9 = 0$. Comparing this to the correct equation $x^2 - 10x + 9 = 0$, the constant term (9) is correct, and only the coefficient of the first-degree term (10 instead of -10) is different. This matches the problem description for student 2.

The determined equation $x^2 - 10x + 9 = 0$ is consistent with the information provided about both students' mistakes and the roots they obtained.

Revision Table: Quadratic Equation Properties

Additional Information: Types of Errors in Quadratic Equations

When solving quadratic equations, making a mistake in a specific coefficient affects the roots in a predictable way. Understanding which property of roots (sum or product) depends on which coefficient is key to solving problems like this.

• Error in the constant term ($c$ or $q$): If the constant term is incorrect, the product of the roots obtained from the incorrect equation will be wrong, but the sum of the roots will be correct (as the coefficient of the first-degree term is assumed to be correct).
• Error in the coefficient of the first-degree term ($b$ or $p$): If the coefficient of the first-degree term is incorrect, the sum of the roots obtained from the incorrect equation will be wrong, but the product of the roots will be correct (as the constant term is assumed to be correct).
• Error in the coefficient of the second-degree term ($a$): If the $a$ coefficient is incorrect (and the equation is solved as $ax^2+bx+c=0$), it would affect both the sum ($-\frac{b}{a}$) and product ($\frac{c}{a}$) of roots unless the other coefficients are changed proportionally. In the standard form $x^2+px+q=0$, if $a$ was the original mistake, it effectively means both $p$ and $q$ are incorrect unless stated otherwise. This problem simplifies things by assuming the standard form $x^2+px+q=0$ where $p$ relates to the first-degree coefficient and $q$ to the constant term, and only one of these is wrong.

By isolating the correct piece of information from each student's attempt, we can reconstruct the correct original quadratic equation.