Question

 If one root of the equation is 2, find the other root of the equation ${{x}^{2}}-5x+6$
A. 5
B. 3
C. 7
D. 8

Answer 1

Hint: Here, one root is given for the quadratic equation ${{x}^{2}}-5x+6=0$. Now, we can find the other root by the formula for sum and product of the roots. If $\alpha$ and $\beta$ are the two roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ then the sum and product of the roots are given by the formula: $\alpha +\beta =\dfrac{-b}{a}$ and $\alpha \beta =\dfrac{c}{a}$.

Complete step-by-step answer:
 Here, given the quadratic equation ${{x}^{2}}-5x+6=0$ and one root of the equation is 2.
Now, we have to find the other root of the equation.
We know that a quadratic equation in the variable $x$ is an equation of the form $a{{x}^{2}}+bx+c=0$ where $a,b,c$ are real numbers $a\ne 0$
We also know that a quadratic equation has two roots. The roots of the equation are given by the formula:
$\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$.
For the quadratic equation $a{{x}^{2}}+bx+c=0$, if $\alpha $ and $\beta $ are the two roots then the sum of the roots is given by the formula:
$\alpha +\beta =\dfrac{-b}{a}$
Hence, the product of the roots is given by the formula:
$\alpha \beta =\dfrac{c}{a}$
Here, corresponding to the quadratic equation $a{{x}^{2}}+bx+c=0$, we have the quadratic equation
${{x}^{2}}-5x+6=0$ where $a=1,b=-5,c=6$
Here, one root is given which is 2. Now to find the other root consider the sum of the roots of the quadratic equation:
$\begin{align}
  & \alpha +\beta =\dfrac{-b}{a} \\
 & 2+\beta =\dfrac{-(-5)}{1} \\
 & 2+\beta =\dfrac{5}{1} \\
 & 2+\beta =5 \\
\end{align}$
Now, by taking 2 to the right side it becomes -2, hence, we get:
$\begin{align}
  & \beta =5-2 \\
 & \beta =3 \\
\end{align}$
Hence, we will get the other root as 3.
Therefore, the two roots of the equation are 2 and 3.
Hence, the correct answer for this question is option (b).

Note: Here, we can also find the roots by directly substituting the values of $a,b$ and $c$ in the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$. Otherwise, you can find the roots by splitting the terms and finding the factors, if you don’t know the formula for sum and product of the roots.