(3)

$D=b^2–4ac=4^2–4\times 2\times 3=16–24=–8<0$

Since $D<0$

Hence, roots are not real.

(2)

Factors of $p=p\times 1$

$\therefore$ $\text{Roots are }p\text{ and }1.$

The quadratic equation is:

$x^2-\text{(sum of roots)}x+\text{product of roots}=0$

$\Rightarrow x^2-(p+1)x+p=0$

(3)

If the discriminant is equal to zero, i.e., $b^2-4ac=0$ where a, b, c are real numbers and $a\ne 0,$ then roots of the quadratic equation $a{x}^2+bx+c=0,$ are real and equal $b^2-4ac=0\Rightarrow ac=\frac{b^2}{4}$

(2)        – 11

(4)

Given equation is $x^2+x+1=0$

Where $a=1,$ $b=1,$ $c=1$

$D=b^2-4ac={\left(1\right)}^2-4\times 1\times 1$

$\Rightarrow D=-3$

Where $D<0$

When $D<0$ roots are not-real.

(1)

Given equation is $x^{2/3} + x^{1/3}- 2 = 0$

⇒ t² + t - 2 = 0     (Take $x^{1/3}$ = t)

⇒ t² + 2t - t - 2 = 0

⇒ t(t + 2) - 1(t + 2) = 0     ⇒ (t + 2)(t - 1) = 0

The roots of above equation are t = 1 and t = -2.

If t = 1, then x = 1.

If t = -2, then x = -8.