Quadratic Equations Class 11 – Consider the quadratic equation: $px^{2}+qx+r=0$ with real coefficients p, q, r and $p\neq 0$. Now, let us assume that the discriminant d < 0 i.e. $b^{2}-4ac< 0$. The solution of above quadratic equation will be in the form of complex numbers given by, $x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}=\frac{-b\pm i\sqrt{4ac-b^{2}}}{2a}$

Note:

1. A polynomial equation has at least one root.
2. A polynomial equation of degree n has n roots.

## Quadratic Equations Class 11 Examples

1. Find the roots of equation $x^{2}+2=0$

Solution: Give, $x^{2}+2=0$

i.e. $x^{2} = -2$ or x = $\pm \sqrt{2}i$

2. Solve $\sqrt{5}x^{2}+x+\sqrt{5}=0$

Solution: Given $\sqrt{5}x^{2}+x+\sqrt{5}=0$

Therefore, discriminant D = $b^{2}-4ac=1-4(\sqrt{5}\times \sqrt{5})=-19$

Therefore, the solution of given quadratic equation = $\frac{-1\pm \sqrt{-19}}{2\sqrt{5}}=\frac{-1\pm 19i}{2\sqrt{5}}$

3. Solve $x^{2}+x+1=0$

Solution: Given $x^{2}+x+1=0$

Therefore, discriminant D = $b^{2}-4ac=1-4=-3$

Therefore, the solution of given quadratic equation = $\frac{-1\pm \sqrt{-3}}{2}=\frac{-1\pm 3i}{2}$