[[1]] and ____ are the roots of 4x2+3x+5=0 by the method of completing the square.

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Detailed Solution

Given quadratic equation is 4x2+3x+5=0
There are different steps to find roots of a given quadratic equation by completing the square method.
In this method we first make the shift constant term to the right hand side if it is on the left side.
4x2+3x=−5
In the second step we remove the coefficient of x2 if it is there or if not then proceed to the third step.
As a given problem, the coefficient of x2 is 4.
So, we remove it or either we make it one.
Divide each term by 4 to make the coefficient of x2 as 1.

\Rightarrow \frac{4 x^{2}}{4} + \frac{3 x}{4} = - \frac{5}{4}

Or

x^{2} + \frac{3}{4} x = - \frac{5}{4}

In the third step we add a square of half of the coefficient of ‘x’ on both sides of the above equation.

\Rightarrow x^{2} + \frac{3}{4} x + {[\frac{1}{2} (\frac{3}{4})]}^{2} = - \frac{5}{4} + {[\frac{1}{2} (\frac{3}{4})]}^{2}

\Rightarrow x^{2} + \frac{3}{4} x + {(\frac{3}{8})}^{2} = - \frac{5}{4} + \frac{9}{64}

\Rightarrow {(x + \frac{3}{8})}^{2} = \frac{- 80 + 9}{64}

\Rightarrow {(x + \frac{3}{8})}^{2} = - \frac{71}{64}

Clearly in the above equation we see that the left hand side is square and the right hand side is a negative. Which can’t be possible.
Hence, from above we can say that the roots of a given quadratic equation is not real.
Roots of the quadratic equation 4x2+3x+5=0 are complex roots.
These complex roots can be found by taking the square root of the right hand side.