 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 = $$\frac{-1\pm \sqrt{-3}}{2}=\frac{-1\pm 3i}{2}$$