Chapter 12, Problem 8CR

To determine

To define: The number of real roots of the equation $x^2+2x+5=0$ and the choose correct option from the following options,

  $\begin{array}{l}\text{a}.0\\ \text{b}\text{.}1\\ \text{c}.2\\ \text{d}\text{.}3\end{array}$

Answer to Problem 8CR

The correct option of equation $x^2+2x+5=0$ is $\text{a}.0$ .

Explanation of Solution

Given information:

The equation $x^2+2x+5=0$ and provided options are,

  $\begin{array}{l}\text{a}.0\\ \text{b}\text{.}1\\ \text{c}.2\\ \text{d}\text{.}3\end{array}$

The provided equation as:

  $x^2+2x+5=0$ .

If $a{z}^2+bz+c=0$ , $a\ne 0$

Then number of real roots are,

If $b^2-4ac>0$ then $a{z}^2+bz+c=0$ have 2 different real roots.

If $b^2-4ac=0$ then $a{z}^2+bz+c=0$ have 2 repeated real roots.

If $b^2-4ac<0$ then $a{z}^2+bz+c=0$ have no real roots.

Using the rule of real roots on discriminant of quadratic equation.

The values of $a=1,b=2\text{and}c=5$ put into the discriminant of quadratic equation,

  $\begin{array}{l}{b}^2-4ac\\ ={\left(2\right)}^2-4\cdot 1\cdot 5\\ =4-20\\ =-16\end{array}$

The discriminant of quadratic equation is negative.

So, by this rule If $b^2-4ac<0$ then $a{z}^2+bz+c=0$ have no real roots, there are no real roots of the equation.