If the equations ${x^2} + bx + c = 0$ have the root of opposite sign of the roots of the equation ${x^2} - 5x + 6 = 0$, then find the value of $b$ and $c$?
A. $b = 5,c = - 6$
B. $b = - 5,c = 6$
C. $b = - 5,c = - 6$
D. None of these

Hint:
We will first find the roots of the given equation, ${x^2} - 5x + 6 = 0$ and then change their sign to write the roots of ${x^2} + bx + c = 0$. Next, we will apply the rule, which is , if $\alpha $ and $\beta $ are roots of a equation, then the equation is of the form ${x^2} - \left( {\alpha + \beta } \right)x + \alpha \beta $, to find the corresponding equation. Next, compare the equation with the given equation to find the value of $b$ and $c$.

Complete step by step solution:
We will first find the roots of the equation ${x^2} - 5x + 6 = 0$
We will factorise the equation ${x^2} - 5x + 6 = 0$
$ {x^2} - 3x - 2x + 6 = 0 \\
   \Rightarrow x\left( {x - 3} \right) - 2\left( {x - 3} \right) = 0 \\
   \Rightarrow \left( {x - 2} \right)\left( {x - 3} \right) = 0 $
Equate each factor to 0 to find the roots of the equation.
$ x - 2 = 0 \\
   \Rightarrow x = 2 $
And
$ x - 3 = 0 \\
   \Rightarrow x = 3 $
If the equation ${x^2} - 5x + 6 = 0$ has roots 2 and 3, then the equation ${x^2} + bx + c = 0$ has roots $ - 2$ and $ - 3$.
Now, we now that if $\alpha $ and $\beta $ are roots of a equation, then the equation is of the form ${x^2} - \left( {\alpha + \beta } \right)x + \alpha \beta $
Therefore, the equation with the roots $ - 2$ and $ - 3$ is ${x^2} - \left( { - 2 - 3} \right)x + \left( { - 2} \right)\left( { - 3} \right) = {x^2} + 5x + 6 = 0$
Hence, the value of $b$ is 5 and the value of $c$ is 6.

Thus, option D is correct.

Note:
If the equation is quadratic, then it can have at most 2 roots. Here, we calculated the roots using factorisation, but roots can also be calculated using the formula $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, where the equation is \[a{x^2} + bx + c = 0\].