Question

Find the zeros of the given quadratic polynomial $4{{x}^{2}}-4x-3$.

Answer 1

Hint: We will simply factorize the given quadratic equation to get the zeros of the quadratic equation. While factorization we will split the middle term $-4x$ such that addition of the new terms will be equal to $-4x$ and multiplication of two new terms formed after splitting will be $4{{x}^{2}}\times \left( -3 \right)=-12{{x}^{2}}$.

Complete step-by-step solution -
It is given in the question that to find the zero of the given quadratic polynomial $4{{x}^{2}}-4x-3$, for this we will factorize the given quadratic polynomial. For factorization, we will split the middle term $-4x$ such that addition of the new terms will be equal to $-4x$ and multiplication of two new terms formed after splitting will be $4{{x}^{2}}\times \left( -3 \right)=-12{{x}^{2}}$.
Now, given polynomial $4{{x}^{2}}-4x-3$, we will split $-4x$ as $-6x+2x$, therefore we get, = $4{{x}^{2}}-6x+2x-3$ Now, we find that the factor $\left( 2x-3 \right)$ can be separated out common from the equation as follows = $2x\left( 2x-3 \right)+1\left( 2x-3 \right)$ simplifying further, we get = $\left( 2x-3 \right)\left( 2x+1 \right)$. Now, to find the zeros of the polynomial expression or to find roots of the equation, we need to equate the 2 factors so obtained to 0. That is, $2x+1=0$ and \[2x-3=0\], therefore we get the values of x or we get roots of the polynomial as \[x=\dfrac{-1}{2}\] and $x=\dfrac{3}{2}$.
Thus, the roots of the given quadratic polynomial are obtained as $\dfrac{-1}{2}\text{ and }\dfrac{3}{2}$.

Note: This is a very basic question of algebra but usually students confuse in splitting the middle term. Many students may take the wrong sign while splitting and which result in the formation of the wrong answer. Thus, it is recommended to split the middle term carefully and avoid silly mistakes while solving this question.