Sachin and rahul attempted to slove a quadratic eqn. Sachin made a mistake in writing down the constant term and ended up it roots (4,3). Rahul made a mistake in writing down coefficient of x to get roots (3,2) the correct roots of the eqn are:
Sachins would have been
(x-4)(x-3) = 0
x^2 - 7x + 12 = 0
Rahul's would have been
(x-3)(x-2) = 0
x^2 - 5x + 6 = 0
Sachin was wrong with the 12, should have been 6
Rahul was wrong with the -5x , should have been -7x
Equation should have been
x^2 - 7x + 6 = 0
(x-6)(x-1) = 0
x = 6 or x = 1
To find the correct roots of the quadratic equation, we need to first determine the equation itself.
Let's assume the quadratic equation in the standard form is: ax^2 + bx + c = 0.
According to the problem, Sachin made a mistake in writing down the constant term, leading to roots (4, 3). This means that when substituting these roots into the equation, it should satisfy the following conditions:
1. Substitute x = 4 into the equation: a(4)^2 + b(4) + c = 0
Simplifying this: 16a + 4b + c = 0
Equation 1: 16a + 4b + c = 0
2. Substitute x = 3 into the equation: a(3)^2 + b(3) + c = 0
Simplifying this: 9a + 3b + c = 0
Equation 2: 9a + 3b + c = 0
Similarly, Rahul made a mistake in writing down the coefficient of x, resulting in roots (3, 2). Therefore, the following conditions should hold true:
3. Substitute x = 3 into the equation: a(3)^2 + b(3) + c = 0
Simplifying this: 9a + 3b + c = 0
Equation 3: 9a + 3b + c = 0
4. Substitute x = 2 into the equation: a(2)^2 + b(2) + c = 0
Simplifying this: 4a + 2b + c = 0
Equation 4: 4a + 2b + c = 0
Now we have four equations (Equations 1, 2, 3, and 4) with three variables (a, b, and c). We can solve this system of equations to find the correct values of a, b, and c.
By subtracting Equation 2 from Equation 1, we eliminate "c":
(16a + 4b + c) - (9a + 3b + c) = 0
Simplifying: 7a + b = 0 --> Equation 5
By subtracting Equation 4 from Equation 3, we eliminate "c":
(9a + 3b + c) - (4a + 2b + c) = 0
Simplifying: 5a + b = 0 --> Equation 6
From Equation 5 and Equation 6, we can observe that both equations have the same form (7a + b = 0 and 5a + b = 0). Subtracting Equation 6 from Equation 5, we can eliminate "b":
(7a + b) - (5a + b) = 0
Simplifying: 2a = 0
Therefore, a = 0
By substituting a = 0 into Equation 5, we can solve for "b":
7(0) + b = 0
Simplifying: b = 0
Now that we have a = 0 and b = 0, we can substitute these values into any of the four equations (Equations 1-4) to solve for "c". Let's use Equation 1:
16(0) + 4(0) + c = 0
Simplifying: c = 0
Therefore, the correct equation is:
0x^2 + 0x + 0 = 0
This equation simplifies to 0 = 0, which means it is an identity and holds true for all real values of x. In other words, the equation has infinite solutions, and it is not a quadratic equation with distinct roots.
To find the correct roots of the quadratic equation, we need to first determine the quadratic equation itself.
Let's assume the quadratic equation is in the standard form: ax^2 + bx + c = 0.
From the information given, we know that both Sachin and Rahul made mistakes. Sachin made a mistake in writing down the constant term, while Rahul made a mistake in writing down the coefficient of x.
Let's consider Sachin's mistake:
He wrote down the roots as (4,3). The roots of a quadratic equation in the form (x - p)(x - q) = 0 are given by (p, q). So, we can set up the equation based on the roots as follows:
(x - 4)(x - 3) = 0
Expanding this equation gives us:
x^2 - 7x + 12 = 0
Now let's consider Rahul's mistake:
He wrote down the roots as (3,2). We can again set up the equation based on the roots as follows:
(x - 3)(x - 2) = 0
Expanding this equation gives us:
x^2 - 5x + 6 = 0
Now, we need to find the correct quadratic equation. We can combine the two equations obtained from Sachin's and Rahul's mistakes to eliminate the errors.
To do this, we can equate the coefficients of both equations:
x^2 - 7x + 12 = x^2 - 5x + 6
By simplifying the equation:
-7x + 12 = -5x + 6
Bringing like terms to one side:
-7x + 5x = 6 - 12
-2x = -6
Dividing both sides by -2:
x = 3
Now, let's substitute this value of x into one of the equations (preferably the simpler one) to find the correct roots.
Using the equation x^2 - 5x + 6 = 0, we substitute x = 3:
3^2 - 5(3) + 6 = 0
9 - 15 + 6 = 0
0 = 0
So, the correct roots of the quadratic equation are (3, 2).