Question

If $5{{x}^{2}}-3x-2=0$, then $x=3,-\dfrac{2}{5}$. Is this if and then the relationship is correct?
A.Yes
B.No
C.Ambiguous
D.Data insufficient

Answer 1

Hint: Solve the given quadratic equation by factorization method. We are factorizing the given quadratic equation by writing the coefficient of x as the difference of factors of 5 and 2 which is $5{{x}^{2}}-\left( 5-2 \right)x-2=0$. Simplification of this expression will give us the factors as $\left( 5x+2 \right)\left( x-1 \right)=0$. Now find the values of x from this equation and then compare with the given values of x.

Complete step-by-step answer:
The quadratic equation given in the question is:
$5{{x}^{2}}-3x-2=0$
We are going to solve the above quadratic equation by factorization method. In factorization method, what we will do is to write the coefficient of x i.e. 3 in the addition or subtraction of the factors of 5 and 2. If we subtract 2 from 5 we will get 3 so we can write the above quadratic equation as follows:
$5{{x}^{2}}-\left( 5-2 \right)x-2=0$
Now, bringing x inside the bracket in the above equation we get,
$5{{x}^{2}}-\left( 5x-2x \right)-2=0$
Opening the bracket in the above equation by multiplying the minus sign we get,
$5{{x}^{2}}-5x+2x-2=0$
If you can see carefully the above equation in the first two terms 5x is common and in the last two terms 2 is common so taking these commons in the above equation we get,
$5x\left( x-1 \right)+2\left( x-1 \right)=0$
As you can see that $\left( x-1 \right)$ is common in the above expression so taking $\left( x-1 \right)$ as common and writing the remaining expression.
$\left( x-1 \right)\left( 5x+2 \right)=0$
Equating $\left( x-1 \right)$ to 0 we get,
$\begin{align}
  & x-1=0 \\
 & \Rightarrow x=1 \\
\end{align}$
Equating $\left( 5x+2 \right)$ to 0 we get,
$\begin{align}
  & 5x+2=0 \\
 & \Rightarrow x=-\dfrac{2}{5} \\
\end{align}$
From the above, we have found the solutions of the given quadratic equation as $x=1\And x=-\dfrac{2}{5}$. On comparing these values of x with the values of x given in the question $x=3,-\dfrac{2}{5}$ we have found that only $x=-\dfrac{2}{5}$ is same as we have derived. So, we can say that if $5{{x}^{2}}-3x-2=0$, then $x=3,-\dfrac{2}{5}$ is not correct.
Hence, the correct option is (b).

Note: There is an alternative way of solving this problem that substitutes the values of x given in the question in the given quadratic equation and then see whether these values of x are satisfying the given quadratic equation or not.
The given quadratic equation:
$5{{x}^{2}}-3x-2=0$
Substituting the value of $x=3$ in the above equation we get,
$\begin{align}
  & 5{{\left( 3 \right)}^{2}}-3\left( 3 \right)-2=0 \\
 & \Rightarrow 5\left( 9 \right)-9-2=0 \\
 & \Rightarrow 45-11=0 \\
 & \Rightarrow 34=0 \\
\end{align}$
As you can see that L.H.S is not equal to R.H.S so $x=3$ is not satisfying the given quadratic equation.