Quadratic Equations Class 11

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