# x^2+10=0 Find the nature of the solutions of the equation.

## Question:

{eq}x^2+10=0 {/eq}

Find the nature of the solutions of the equation.

## Nature of Solutions of a Quadratic Equation

A mathematical equation is called a quadratic equation if it has degree 2 means the highest power of a variable in a quadratic equation is equal to 2.

general form-

{eq}\displaystyle px^{2}+qx+r = 0 , p \neq 0 {/eq}

here p,q, and r are the constants

x is a variable

A quadratic equation has two solutions that can be real or imaginary.

Imaginary solutions of the quadratic equation are also called complex solutions.

The solutions of a quadratic equation {eq}\displaystyle px^{2}+qx+r = 0 {/eq} are given by using the quadratic formula written below-

{eq}\displaystyle x = \frac{-q \pm \sqrt{q^{2}-4pr}}{2p} {/eq}

In the above equation, the value of {eq}\displaystyle q^{2}-4pr {/eq} determines the nature of solutions.

{eq}q^{2}-4pr {/eq} is called the discriminant (D) of the quadratic equation.

{eq}D = q^{2}-4pr {/eq}

#### Nature of the Solutions

(1) If {eq}D > 0 {/eq} then the solutions of the quadratic equation will be real and distinct.

(2) If {eq}D < 0 {/eq} then the solutions of the quadratic equation will be imaginary or complex. In this situation, there is no real solution to the equation.

(1) If {eq}D = 0 {/eq} then the solutions of the quadratic equation will be real and equal.

{eq}x^{2}+10 = 0 --------(1) {/eq}

Now on comparing the equation(1) with {eq}px^{2}+qx+r = 0 {/eq} we have

{eq}p = 1 {/eq}

{eq}q = 0 {/eq}

{eq}r = 10 {/eq}

So for finding the nature of the roots, we have to calculate the discriminant.

{eq}D = (0)^{2}-4 \times 1\times 10 {/eq}

{eq}D = -40 < 0 {/eq}

So the given quadratic equation will have distinct imaginary or complex solutions.