Question

The roots of the equation \[{x^2} - 3x - 10 = 0\] are
(A) Roots are $1$ and $3.$
(B) Roots are $ - 2$ and $5.$
(C) Roots are $ - 8$ and $2.$
(D) Roots are $4$ and $3.$

Answer 1

Hint: In this question first we will multiply the constant term and the coefficient of ${x^2}$ and then we will find the factors of the result. Now, we will try to add or subtract the factors to form the coefficient of $x$ . Now, we will write the middle term of the equation in the form of the factors we got. Now, take common terms out from the equation we got.

Complete step by step solution: The given equation is \[{x^2} - 3x - 10 = 0\]
To find the roots of \[{x^2} - 3x - 10 = 0.\] We will the coefficient of ${x^2}$ i.e. $1$ and the constant term $10$.
Now we will find the factors of $10$. The factors of $10$ are $5$ and $2$ . Now, we will write the middle term of the equation as the difference of $5$ and $2$ i.e. we will write $ - 3x$ as $ - 5x + 2x$.
Therefore, the equation \[{x^2} - 3x - 10 = 0\] can be written as follows:
\[{x^2} - 3x - 10 = 0\]
$ \Rightarrow {x^{^2}}-5x+2x-10=0$
Take common terms out from the above equation
\[ \Rightarrow x\left( {x - 5} \right) + 2\left( {x - 5} \right) = 0\]
Take $\left( {x - 5} \right)$ common from the above equation. Therefore, the above equation can be written as follows:
\[ \Rightarrow \left( {x - 5} \right)\left( {x + 2} \right) = 0\]
Now, we can write,
\[ \Rightarrow \left( {x - 5} \right) = 0\,\,{\text{or}}\,\,\left( {x + 2} \right) = 0\]
Now, by solving the above equation we got the values of $x$.
\[ \Rightarrow x = 5\] or \[x = - 2\]
Therefore, the roots of the equation are $ - 2$ and $5$.
Hence, the option (B) is correct.

Note: In this question be careful while manipulating the factors to form the middle term because it could lead to wrong answers. And also be careful while finding the factors because finding factors could take time. So, just keep practicing these types of questions to become more comfortable in finding the factors.