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By using the method of completing the square, show that the equation 2 x^{2}+x+4=0 has no real roots.
CBSE | Class 10 | Excercise 10B | Maths | Quadratic Equations | RS Aggarwal
By using the method of completing the square, show that the equation 2 x^{2}+x+4=0 has no real roots.

2 x^{2}+x+4=0 \Rightarrow 4 x^{2}+2 x+8=0 \quad (Multiplying both sides by 2)
\Rightarrow 4 x^{2}+2 x=-8
\Rightarrow(2 x)^{2}+2 \times 2 x \times \frac{1}{2}+\left(\frac{1}{2}\right)^{2}=-8+\left(\frac{1}{2}\right)^{2} \quad [Adding \left(\frac{1}{2}\right)^{2} on both sides ]
\Rightarrow\left(2 x+\frac{1}{2}\right)^{2}=-8+\frac{1}{4}=-\frac{31}{4}<0

But, \left(2 x+\frac{1}{2}\right)^{2} cannot be negative for any real value of x.

So, there is no real value of x satisfying the given equation.
Hence, the given equation has no real roots.

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