Question

Consider the given expression \[4{{y}^{2}}+4y+1\ =\ 0\], then \[y\ =\ 0\].
(A). Yes
(B). No
(C). Ambiguous
(D). Data insufficient

Answer 1

- Hint: We can do it by factorizing with the middle term by factorizing method and from that, we will get the value of y then compare with the value of y given and write the correct option.

Complete step-by-step solution -
In the question we are given a quadratic equation \[4{{y}^{2}}+4y+1\ =\ 0\] and then we are given \[y\ =\ 0\] and we have to find \[y\ =\ 0\] satisfy or not.
If the quadratic equation is \[a{{y}^{2}}+by+c\ =\ 0\]
Then \[y\ =\ \dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
Where a is coefficient of \[{{y}^{2}}\], b is coefficient of \[y\] and c is constant.
In the equation \[4{{y}^{2}}+4y+1\ =\ 0\] can be considered as quadratic equation which can be rearranged in standard form as \[a{{y}^{2}}+by+c\ =\ 0\].
Here $y$ represents unknown and $a,b,c$ are known numbers where $a\ne 0$, otherwise it becomes a linear equation.
Consider the given expression \[4{{y}^{2}}+4y+1\ =\ 0\], this can be rewritten as
\[4{{y}^{2}}+2y+2y+1\ =\ 0\]
On factorizing we get,
\[\begin{align}
  & \Rightarrow 2y\left( 2y+1 \right)+1\left( 2y+1 \right)=0 \\
 & \left( 2y+1 \right)\left( 2y+1 \right)\ =\ 0 \\
\end{align}\]
So, the value of \[y\ =\ -\dfrac{1}{2}\], but the value of y given as 0 so, the answer is NO.
So, the correct option is ‘B’.

Note: Another approach is - we can also get the answer by just putting the value of \[y\ =\ 0\] in equation \[4{{y}^{2}}+4y+1\ =\ 0\] to find the answer whether it satisfies or not.