Question

The common root of $$x^2+5x+6=0\text{ and }{x}^2-8x+15=0$$ is the root of $$x^2+4x+q=0$$ then value of q is
A. 21
B. 41
C. -21
D. -41

Answer 1

Hint: At first, find the roots of quadratic equations individually by using the factorisation method. Then, we will find the common root which is also said to be the root of the equation $$x^2+4x+q=0$$. So, to find the value 'q' put the value of 'x' as the common root to get the answer.

Complete step-by-step answer:
In the question, we have been given two quadratic equations $$x^2+5x+6=0\text{ and }{x}^2-8x+15=0$$. Further, we are said that, the common root of the two quadratic equations is also a root of equation $$x^2+4x+q=0.$$
We will first find the roots of the respective quadratic equations $$x^2-5x+6=0\text{ and }{x}^2-8x+15=0$$. We will be using the factorisation method to get the roots as explained below.
So, for the equation $$x^2-5x+6=0$$ we can rewrite it as $$x^2-2x-3x+6=0$$ which can be also written as $$x\left(x-2\right)-3\left(x-2\right)=0$$ and hence factorized as$$\left(x-3\right)\left(x-2\right)=0$$. Thus, the roots of equations are 2 and 3.
Now, we will find for the equation $$x^2-8x+15=0$$ which we can rewrite it as $$x^2-3x-5x+15=0$$ which can also be written as $$x\left(x-3\right)-5\left(x-3\right)=0$$and hence can be factorized as $$\left(x-5\right)\left(x-3\right)=0.$$ Thus, the roots of equation are 5 and 3.
The roots of equation $$x^2-5x+6=0$$ is 2 and 3 and roots of equation $$x^2-8x+15=0$$ is 5 and 3, thus, by observing we can say that, the common root between them is 3.
We were said that, the common root of the equations $$x^2-5x+6=0\text{ and }{x}^2-8x+15=0$$ is also a root of $$x^2+4x+q=0.$$
The common root of $$x^2-5x+6=0\text{ and }{x}^2-8x+15=0$$ is 3, thus, we can say 3 is a root of $$x^2+4x+q=0$$ or we can say that 3 satisfies the quadratic equation $$x^2+4x+q=0.$$
So, now to find the value of 'q' we will substitute x as 3, so we get,
$$\begin{array}{l}{\left(3\right)}^2+4\times 3+q=0\\ \Rightarrow 9+12+q=0\end{array}$$
Now on simplifying, we get,
$$21+q=0\Rightarrow q=-21$$
Hence, the correct option is C.

Note: We can also find the roots of a quadratic equation by using the formula, which is,
$$x={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$$
Where, the quadratic equation is $$a{x}^2+bx+c=0.$$ But this might be confusing and some students might get the formula wrong, so for simple quadratic equations like the ones in this question, the factorisation method is better.