nestalno4szl

2023-02-20

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

Leonard Drake

Beginner2023-02-21Added 6 answers

Step 1: Form the equation

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

We are aware that a polynomial's zeroes are evaluated by equating them to zero.

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

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

Step 2: Solve to find the zeroes

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

$\Rightarrow $$4{x}^{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, $\frac{c}{a}=-\frac{3}{4}$

As a result, the polynomial's zeroes and coefficients are proven to be related.

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

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

We are aware that a polynomial's zeroes are evaluated by equating them to zero.

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

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

Step 2: Solve to find the zeroes

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

$\Rightarrow $$4{x}^{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, $\frac{c}{a}=-\frac{3}{4}$

As a result, the polynomial's zeroes and coefficients are proven to be related.

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