Question

Which two values of x are the roots of the polynomial below? x^2+5x+11

Answer 1

Answer:

x = -5/2 + i√19 and x = -5/2 - i√19

Step-by-step explanation:

Next time, please share the possible answer choices.

Here we can actually find the roots, using the quadratic formula or some other approach.

a = 1, b = 5 and c = 11.  Then the discriminant is b^2-4ac, or 5^2-4(1)(11).  Since the discriminant is negative, the roots are complex.  The discriminant value is 25-44, or -19.

Thus, the roots of the given poly are:

      -5 plus or minus i√19

x = -----------------------------------

                     2(1)

or x = -5/2 + i√19 and x = -5/2 - i√19

Answer 2

Answer:  The roots of the given polynomial are

$x=\dfrac{-5+i\sqrt{19}}{2},~~\dfrac{-5-i\sqrt{19}}{2}.$

Step-by-step explanation:  We are given to find the two values of x that are the roots of the following quadratic polynomial:

$P(x)=x^2+5x+11.$

To find the roots, we must have

$P(x)=0\\\\\Rightarrow x^2+5x+11=0~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

We know that

the solution set of quadratic equation of the form $ax^2+bx+c=0,~a\neq 0$ is given by

$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}.$

From equation (i), we have

a = 1,  b = 5   and   c = 11.

Therefore, the solution of equation (i) is given by

$x\\\\\\=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\\\=\dfrac{-5\pm\sqrt{5^2-4\times1\times 11}}{2\times 1}\\\\\\=\dfrac{-5\pm\sqrt{25-44}}{2}\\\\\\=\dfrac{-5\pm\sqrt{-19}}{2}\\\\\\=\dfrac{-5\pm i\sqrt{19}}{2}.$

Thus, the roots of the given polynomial are

$x=\dfrac{-5+i\sqrt{19}}{2},~~\dfrac{-5-i\sqrt{19}}{2}.$