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Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: \quad x^{2}-4 x-1=0
CBSE | Class 10 | Exercise 10C | Maths | Quadratic Equations | RS Aggarwal
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: \quad x^{2}-4 x-1=0

Given:

x^{2}-4 x-1=0

On comparing it with a x^{2}+b x+c=0, we get: a=1, b=-4 and c=-1

Discriminant D is given by:

\begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =(-4)^{2}-4 \times 1 \times(-1) \\ =16+4 \\ =20 \\ =20>0 \end{array}

Hence, the roots of the equation are real.

Roots \alpha and \beta are given by:

\begin{array}{l} \alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-(-4)+\sqrt{20}}{2 \times 1}=\frac{4+2 \sqrt{5}}{2}=\frac{2(2+\sqrt{5})}{2}=(2+\sqrt{5}) \\ \beta=\frac{-b-\sqrt{D}}{2 a}=\frac{-(-4)-\sqrt{20}}{2}=\frac{4-2 \sqrt{5}}{2}=\frac{2(2-\sqrt{5})}{2}=(2-\sqrt{5}) \end{array}

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