Question

Find the zeroes of the quadratic polynomial $4x^2-4x-3$ and verify the relationship between the zeroes and the coefficients of the polynomial.

Answer 1

Step 1: Form the equation

The given polynomial is $4x^2-4x-3$

We know that the zeroes of a polynomial are evaluated by equating them with zero.

$\therefore p\left(x\right)=0$

$\Rightarrow 4x^2-4x-3=0$

Step 2: Solve to find the zeroes

$4x^2-4x-3=0$

$\Rightarrow$ $4x^2-6x+2x-3=0$

$\Rightarrow$$2x\left(2x-3\right)+1\left(2x-3\right)=0$

$\Rightarrow$ $\left(2x-3\right)\left(2x+1\right)=0$

$\Rightarrow$ $x=\frac{3}{2},-\frac{1}{2}$

Step 3: Verification

We know that for a given polynomial $a{x}^2+bx+c$,

Sum of the zeroes$=-\frac{b}{a}$ and product of the roots$=\frac{c}{a}$

Sum of the zeroes $=\frac{3}{2}-\frac{1}{2}=1$

Again, $-\frac{b}{a}=\frac{4}{4}=1$

Product of the zeroes $=\frac{3}{2}\times -\frac{1}{2}=-\frac{3}{4}$

Again, $\frac{c}{a}=-\frac{3}{4}$

Thus, the relationship between the zeroes and the coefficients of the polynomial is verified.

Hence, the zeroes of this polynomial are $\frac{3}{2}$ and $-\frac{1}{2}$.