Answer
Verified
481.2k+ views
Hint: Compare the equation with the general quadratic equation, then use the formula for finding out the roots of a quadratic equation.
The given quadratic equation is $3{x^2} - 4x + \frac{{20}}{3} = 0$,
Comparing it with general quadratic equation, $a{x^2} + bx + c = 0$, we have:
$a = 3,b = - 4$ and $c = \frac{{20}}{3}$
And we know that the roots of quadratic equation is given as:
$\alpha ,\beta = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Substituting the values from the above equation, we have:
$
\Rightarrow \alpha ,\beta = \frac{{ - ( - 4) \pm \sqrt {{{( - 4)}^2} - 4 \times 3 \times \frac{{20}}{3}} }}{{2(3)}}, \\
\Rightarrow \alpha ,\beta = \frac{{4 \pm \sqrt {16 - 80} }}{6} = \frac{{4 \pm \sqrt { - 64} }}{6}, \\
\Rightarrow \alpha ,\beta = \frac{{4 \pm 8i}}{6} = \frac{{2 \pm 4i}}{3} \\
$
$ \Rightarrow \alpha = \frac{2}{3} + \frac{4}{3}i$ and $\beta = \frac{2}{3} - \frac{4}{3}i.$
Thus the roots of the equation are $\frac{2}{3} + \frac{4}{3}i$ and $\frac{2}{3} - \frac{4}{3}i$
Note: Discriminant of a quadratic equation is:
$ \Rightarrow D = {b^2} - 4ac$
If the discriminant of a quadratic equation is less than zero (i.e. negative), the roots of the equation will always be imaginary and they will be complex conjugates of each other.
The given quadratic equation is $3{x^2} - 4x + \frac{{20}}{3} = 0$,
Comparing it with general quadratic equation, $a{x^2} + bx + c = 0$, we have:
$a = 3,b = - 4$ and $c = \frac{{20}}{3}$
And we know that the roots of quadratic equation is given as:
$\alpha ,\beta = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Substituting the values from the above equation, we have:
$
\Rightarrow \alpha ,\beta = \frac{{ - ( - 4) \pm \sqrt {{{( - 4)}^2} - 4 \times 3 \times \frac{{20}}{3}} }}{{2(3)}}, \\
\Rightarrow \alpha ,\beta = \frac{{4 \pm \sqrt {16 - 80} }}{6} = \frac{{4 \pm \sqrt { - 64} }}{6}, \\
\Rightarrow \alpha ,\beta = \frac{{4 \pm 8i}}{6} = \frac{{2 \pm 4i}}{3} \\
$
$ \Rightarrow \alpha = \frac{2}{3} + \frac{4}{3}i$ and $\beta = \frac{2}{3} - \frac{4}{3}i.$
Thus the roots of the equation are $\frac{2}{3} + \frac{4}{3}i$ and $\frac{2}{3} - \frac{4}{3}i$
Note: Discriminant of a quadratic equation is:
$ \Rightarrow D = {b^2} - 4ac$
If the discriminant of a quadratic equation is less than zero (i.e. negative), the roots of the equation will always be imaginary and they will be complex conjugates of each other.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE