In solving a problem, one student makes a mistake in the coefficient of the first degree term and obtains $ - 9$ and $ - 1$ for the roots. Another student makes a mistake in the constant term of the equation and obtains 8 and 2 for the roots. The correct equation was
(a) ${x^2} + 10x + 9 = 0$
(b) ${x^2} - 10x + 16 = 0$
(c) ${x^2} - 10x + 9 = 0$
(d) None of the above

Hint:
We will first let the correct equation as ${x^2} + bx + c = 0$. Now, let the equation taken by first student is ${x^2} + dx + c = 0$, where student made mistake in first degree term. Similarly, let the equation taken by second is ${x^2} + bx + e = 0$, where student made mistake in the constant term. Now use the property of roots that if the equation is $A{x^2} + Bx + C = 0$, then the sum of roots is $ - B$ and the product of roots of equation is $C$.

Complete step by step solution:
We have to find the correct quadratic equation.
Let the correct quadratic equation be ${x^2} + bx + c = 0$
We are given that the student makes make in the first-degree term of the quadratic equation. Let the student used the first-degree term as $d$
Then, the quadratic equation used by the student is ${x^2} + dx + c = 0$
We are given that the roots calculated by the students are $ - 9$ and $ - 1$.
Also, if the equation is $A{x^2} + Bx + C = 0$, then the sum of roots is $ - B$ and the product of roots of equation is $C$.
Then, according the equation used by student, we have the sum of roots as,
$
   - d = \left( { - 1 + - 9} \right) \\
   \Rightarrow - d = - 10 \\
   \Rightarrow d = 10 \\
$
And the product of roots as,
$
  c = \left( { - 1} \right)\left( { - 9} \right) \\
   \Rightarrow c = 9 \\
$
Now, another student made a mistake in the constant term of the equation.
Let the equation taken the other student be ${x^2} + bx + e = 0$
And the roots calculated were 8 and 2.
The sum of roots of the above equation will be given as,
$
  8 + 2 = - b \\
   \Rightarrow b = - 10 \\
$
And the product of roots is given as
$
  8\left( 2 \right) = e \\
   \Rightarrow e = 16 \\
$
On substituting the value of $b$ and $c$ in the correct equation, we will get,
${x^2} - 10x + 9 = 0$

Hence, option (c) is the correct option.

Note:
In any quadratic equation, the coefficient of ${x^2}$ should not be 0. A quadratic equation can have at most two roots. Also, the sum of the roots is the negative of the coefficient of the first-degree term and the product of the roots is equal to the constant term of the quadratic equation.