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.

### Answer & Explanation

Leonard Drake

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.
$\therefore p\left(x\right)=0$
$⇒4{x}^{2}-4x-3=0$
Step 2: Solve to find the zeroes
$4{x}^{2}-4x-3=0$
$⇒$$4{x}^{2}-6x+2x-3=0$
$⇒$$2x\left(2x-3\right)+1\left(2x-3\right)=0$
$⇒$$\left(2x-3\right)\left(2x+1\right)=0$
$⇒$$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 $=\frac{3}{2}×-\frac{1}{2}=-\frac{3}{4}$
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 $-\frac{1}{2}$.

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