Question

How do you identify the value of the discriminant for the equation $6{x^2} = 2x - 1$?

Answer 1

Hint: In this question we have to find the value of the discriminant we will do this by using discriminant formula and ${b^2} - 4ac$ is called the discriminant, and the nature of the roots can be determined by using discriminant, and by substituting the values in the discriminant we will get the required result.

Complete step by step answer:
Now the given quadratic equation is,
$6{x^2} = 2x - 1$,
Now taking all the terms to one side, we get,
$ \Rightarrow 6{x^2} - 2x + 1 = 0$,
As the equation has a degree 2 we will have two roots for the equation.
Now we will determine the discriminant using the formula which is given by ${b^2} - 4ac$, and the nature roots of any equation can be determined by using discriminant, and if the discriminant i.e., ${b^2} - 4ac$ is greater than zero then the equation will have real roots, if the discriminant i.e., ${b^2} - 4ac$ is equal to zero then the equation will have one real repeated root, if the discriminant i.e., ${b^2} - 4ac$ is less than zero then the equation will have complex roots.
So, here $a = 6$,$b = - 2$,$c = 1$,
Now substituting the values in the discriminant i.e., ${b^2} - 4ac$ we get,
$ \Rightarrow {b^2} - 4ac = {\left( { - 2} \right)^2} - 4\left( 6 \right)\left( 1 \right)$,
Now simplifying we get,
$ \Rightarrow {b^2} - 4ac = 4 - 24$,
Now simplifying we get,
$ \Rightarrow {b^2} - 4ac = - 20$,
So,${b^2} - 4ac = - 20 < 0$, according to the nature of the roots, the given equation will have two complex roots.
Now using the quadratic formula, here it is given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$,
Here $a = 6$,$b = - 2$,$c = 1$,
Now substituting the values in the formula we get,
$ \Rightarrow x = \dfrac{{ - \left( { - 2} \right) \pm \sqrt {{{\left( { - 2} \right)}^2} - 4\left( 6 \right)\left( 1 \right)} }}{{2\left( 6 \right)}}$,
Now simplifying we get,
$ \Rightarrow x = \dfrac{{2 \pm \sqrt {4 - 24} }}{{12}}$,
Now again simplifying we get,
$ \Rightarrow x = \dfrac{{2 \pm \sqrt { - 20} }}{{12}}$,
Now we have negative number inside a square root, we will use the fact that ${i^2} = - 1$, we get,
$ \Rightarrow x = \dfrac{{2 \pm \sqrt {{i^2} \times 20} }}{{12}}$,
Now simplifying we get,
$ \Rightarrow x = \dfrac{{2 \pm \sqrt {4 \times 5 \times {i^2}} }}{{12}}$,
Now taking out the common term we get,
$ \Rightarrow x = \dfrac{{1 \pm i\sqrt 5 }}{6}$
Now we get two values of $x$ they are $\dfrac{{1 + i\sqrt 5 }}{6}$ and $\dfrac{{1 - i\sqrt 5 }}{6}$.
So the zeros of the equation are $\dfrac{{1 \pm \sqrt 5 i}}{6}$, as the discriminant is greater than zero the given equation has two real roots.

$\therefore $ The discriminant for the given equation, i.e., $6{x^2} = 2x - 1$, is $\sqrt { - 20} $ and the zeros of the equation will be $\dfrac{{1 \pm \sqrt 5 i}}{6}$, and the equation will have two complex roots.

Note: The discriminant is part of the quadratic formula which lies underneath the square root. The quadratic equation discriminant is important because it tells us the number and type of solutions. This information is helpful because it serves as a double check when solving quadratic equations by any of the four methods i.e., factoring, completing the square, using square roots, and using the quadratic formula. If you are trying to determine the "type" of roots of any equation we need not complete the entire quadratic formula. Simply look at the discriminant.