Question

Prove that the polynomial x2 + 2x + 5 has no zero

Answer 1

Solution is attached below in image

Answer 2

Step-by-step explanation:

Let $f(x)=x^{2}+2 x+5$

For finding roots to the polynomial, we need to equate it to 0.

$F(x) = 0$

$x^{2}+2 x+5=0$

For a quadratic polynomial a x^{2}+b x+c=0 to have roots, the discriminant needs to be examined.

i.e., If $b^{2}-4 a c>0$, the roots are real.

If $b^{2}-4 a c<0$, the roots are imaginary.

If $b^{2}-4 a c=0$, the roots are equal.

In the given problem, $x^{2}+2 x+5=0$ in which a=1, b=2, c=5

$b^{2}-4 a c=2^{2}-4(1)(5)=4-20=-16<0$

The roots are imaginary

There are no real roots for the given quadratic equation.

Hence, proved.