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Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: x^{2}-6 x+4=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: x^{2}-6 x+4=0

Given:

x^{2}-6 x+4=0

On comparing it with a x^{2}+b x+c=0, we get: a=1, b=-6 and c=4

Discriminant D is given by:

\begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =(-6)^{2}-4 \times 1 \times 4 \\ =36-16 \\ =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{-(-6)+\sqrt{20}}{2 \times 1}=\frac{6+2 \sqrt{5}}{2}=\frac{2(3+\sqrt{3})}{2}=(3+\sqrt{5}) \\ \beta=\frac{-b-\sqrt{D}}{2 a}=\frac{-(-6)-\sqrt{20}}{2}=\frac{6-2 \sqrt{5}}{2}=\frac{2(3-\sqrt{5})}{2}=(3-\sqrt{5}) \end{array}

Thus, the roots of the equation are (3+2 \sqrt{5}) and (3-2 \sqrt{5}).

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