Answer
Verified
488.4k+ views
Hint- Here, we will be using a discriminant method to solve the given quadratic equation.
Given, equation is $\sqrt 3 {x^2} - 2\sqrt 2 x - 2\sqrt 3 = 0$
As we know that for any general quadratic equation $a{x^2} + bx + c = 0$, the solution is given as
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ where $d = \sqrt {{b^2} - 4ac} $ is the discriminant of the quadratic equation.
On comparing the given quadratic equation with the general quadratic equation, we get
$a = \sqrt 3 $ ,$b = - 2\sqrt 2 $ and $c = - 2\sqrt 3 $
Now substitute these values in the formula, we get
$
x = \dfrac{{ - \left( { - 2\sqrt 2 } \right) \pm \sqrt {{{\left( { - 2\sqrt 2 } \right)}^2} - 4\left( {\sqrt 3 } \right)\left( { - 2\sqrt 3 } \right)} }}{{2\left( {\sqrt 3 } \right)}} = \dfrac{{2\sqrt 2 \pm \sqrt {8 + 24} }}{{2\sqrt 3 }} = \dfrac{{2\sqrt 2 \pm \sqrt {32} }}{{2\sqrt 3 }} \\
\Rightarrow x = = \dfrac{{2\sqrt 2 \pm 4\sqrt 2 }}{{2\sqrt 3 }} = \dfrac{{\sqrt 2 \pm 2\sqrt 2 }}{{\sqrt 3 }} \\
$
$ \Rightarrow {x_1} = \dfrac{{\sqrt 2 + 2\sqrt 2 }}{{\sqrt 3 }} = \dfrac{{3\sqrt 2 }}{{\sqrt 3 }} = \left( {\sqrt 3 } \right)\left( {\sqrt 2 } \right) = \sqrt 6 $ and $ \Rightarrow {x_2} = \dfrac{{\sqrt 2 - 2\sqrt 2 }}{{\sqrt 3 }} = \dfrac{{ - \sqrt 2 }}{{\sqrt 3 }} = - \sqrt {\dfrac{2}{3}} $ .
i.e., The two roots of the given quadratic equation are ${x_1} = \sqrt 6 $ and ${x_2} = - \sqrt {\dfrac{2}{3}} $.
Therefore, the two values of $x$ possible in order to satisfy the given quadratic equations are $\sqrt 6 $ and $ - \sqrt {\dfrac{2}{3}} $.
Note- For any quadratic equation, $a{x^2} + bx + c = 0$, according to the value of $d = \sqrt {{b^2} - 4ac} $ we have three possible cases:
i. If it is positive, then the quadratic equation will have two different real roots.
ii. If it is equal to zero, then the quadratic equation will have real and equal roots.
iii. If it is negative, then the quadratic equation will have two different imaginary roots.
Given, equation is $\sqrt 3 {x^2} - 2\sqrt 2 x - 2\sqrt 3 = 0$
As we know that for any general quadratic equation $a{x^2} + bx + c = 0$, the solution is given as
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ where $d = \sqrt {{b^2} - 4ac} $ is the discriminant of the quadratic equation.
On comparing the given quadratic equation with the general quadratic equation, we get
$a = \sqrt 3 $ ,$b = - 2\sqrt 2 $ and $c = - 2\sqrt 3 $
Now substitute these values in the formula, we get
$
x = \dfrac{{ - \left( { - 2\sqrt 2 } \right) \pm \sqrt {{{\left( { - 2\sqrt 2 } \right)}^2} - 4\left( {\sqrt 3 } \right)\left( { - 2\sqrt 3 } \right)} }}{{2\left( {\sqrt 3 } \right)}} = \dfrac{{2\sqrt 2 \pm \sqrt {8 + 24} }}{{2\sqrt 3 }} = \dfrac{{2\sqrt 2 \pm \sqrt {32} }}{{2\sqrt 3 }} \\
\Rightarrow x = = \dfrac{{2\sqrt 2 \pm 4\sqrt 2 }}{{2\sqrt 3 }} = \dfrac{{\sqrt 2 \pm 2\sqrt 2 }}{{\sqrt 3 }} \\
$
$ \Rightarrow {x_1} = \dfrac{{\sqrt 2 + 2\sqrt 2 }}{{\sqrt 3 }} = \dfrac{{3\sqrt 2 }}{{\sqrt 3 }} = \left( {\sqrt 3 } \right)\left( {\sqrt 2 } \right) = \sqrt 6 $ and $ \Rightarrow {x_2} = \dfrac{{\sqrt 2 - 2\sqrt 2 }}{{\sqrt 3 }} = \dfrac{{ - \sqrt 2 }}{{\sqrt 3 }} = - \sqrt {\dfrac{2}{3}} $ .
i.e., The two roots of the given quadratic equation are ${x_1} = \sqrt 6 $ and ${x_2} = - \sqrt {\dfrac{2}{3}} $.
Therefore, the two values of $x$ possible in order to satisfy the given quadratic equations are $\sqrt 6 $ and $ - \sqrt {\dfrac{2}{3}} $.
Note- For any quadratic equation, $a{x^2} + bx + c = 0$, according to the value of $d = \sqrt {{b^2} - 4ac} $ we have three possible cases:
i. If it is positive, then the quadratic equation will have two different real roots.
ii. If it is equal to zero, then the quadratic equation will have real and equal roots.
iii. If it is negative, then the quadratic equation will have two different imaginary roots.
Recently Updated Pages
what is the correct chronological order of the following class 10 social science CBSE
Which of the following was not the actual cause for class 10 social science CBSE
Which of the following statements is not correct A class 10 social science CBSE
Which of the following leaders was not present in the class 10 social science CBSE
Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE
Which one of the following places is not covered by class 10 social science CBSE
Trending doubts
A rainbow has circular shape because A The earth is class 11 physics CBSE
Which are the Top 10 Largest Countries of the World?
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Why is there a time difference of about 5 hours between class 10 social science CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell