In the following, determine whether the given quadratic equations have real roots and if so, find the roots:

$3x^2-5x+2=0$

Given:

Given quadratic equation is $3x^2-5x+2=0$.

To do:

We have to determine whether the given quadratic equation has real roots.

Solution:

Comparing the given quadratic equation with the standard form of the quadratic equation $ax^2+bx+c=0$, we get,

$a=3, b=-5$ and $c=2$.

The discriminant of the standard form of the quadratic equation $ax^2+bx+c=0$ is

$D=b^2-4ac$.

Therefore,

$D=(-5)^2-4(3)(2)=25-24=1$.

As $D>0$, the given quadratic equation has real roots and the roots are

$x=\frac{-b\pm \sqrt{D}}{2a}$

$x=\frac{-(-5)\pm \sqrt{1}}{2(3)}$ 

$x=\frac{5\pm 1}{6}$ 

$x=\frac{5+1}{6}$ or $x=\frac{5-1}{6}$

$x=\frac{6}{6}$ or $x=\frac{4}{6}$

$x=1$ or $x=\frac{2}{3}$

The roots are $1$ and $\frac{2}{3}$. 

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Updated on: 10-Oct-2022

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