Coefficients Of X

Coefficients Of X - Quadratic Equation

Question:

In copying a quadratic equation of the form x2 + px + q = 0, a student wrote the coefficients of x incorrectly and the roots were found to be 3 and 10; another student wrote the same equation but he wrote the constant term incorrectly and thus, he found the roots to be 4 and 7. The roots of the correct question are:

Solution:

Quadratic equation in the conception of algebra can be represented as:
ax2 + bx + c = 0
In this equation x is considered unknown, ‘a’ and ‘b’ are the coefficient of x and c is the constant number. The roots of a quadratic equation can be extracted by two ways, firstly by simplifying the equation and secondly, by the standard formula:
x = {- b ± (√b2 – 4ac}/2a
Quadratic equation mentioned in the question is x2 + px+ q 
If the roots given in the question are multiplied then the result will be: 
3 x 10 = 30 
This value represents the value of q
Therefore, q = 30
As per the second student, the value of the roots found is 4 and 7 (given in the problem)
If the roots will be added, then the value of p can be extracted. 
Adding 4 and 7 results in 11
Thus, -p will be equal to 11
=> p = - 11 
The extracted values of q and p will contribute towards the extraction of the new or correct quadratic equation,
x2 – 11x +30 = 0
Simplifying the correct quadratic equation for calculating x
=> x2 – 6x – 5x + 30 = 0 
=> x(x – 6) – 5(x – 6) = 0
=> (x – 5) (x – 6) = 0
Thus, the roots of the new quadratic equation are 5 and 6
Therefore, 5 and 6 are the new or correct roots of the correct quadratic equation, which is x2 – 11x +30 = 0

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