Answer
Verified
484.2k+ views
Hint: Here, we will be putting the discriminant of the given quadratic equation equal to zero in order for this equation to have two equal roots. From here we will obtain the value of unknown k.
Complete step-by-step answer:
The given quadratic equation is \[
kx\left( {x - 2} \right) + 6 = 0 \\
\Rightarrow k{x^2} - 2kx + 6 = 0{\text{ }} \to {\text{(1)}} \\
\]
As we know that for any general quadratic equation \[a{x^2} + bx + c = 0{\text{ }} \to {\text{(2)}}\] to have two equal roots, the discriminant should be equal to zero where the discriminant is given by
${\text{d}} = \sqrt {{b^2} - 4ac} \to {\text{(3)}}$
By comparing equations (1) and (2), we get
a=k, b=-2k and c=6
It is also given that the given quadratic equation given by equation (1) has two equal roots.
So, d=0 for the given quadratic equation.
$
\Rightarrow \sqrt {{{\left( { - 2k} \right)}^2} - 4\left( k \right)\left( 6 \right)} = 0 \\
\Rightarrow \sqrt {4{k^2} - 24k} = 0 \\
\Rightarrow 4{k^2} - 24k = 0 \\
\Rightarrow 4k\left( {k - 6} \right) = 0 \\
$
Either $
4k = 0 \\
\Rightarrow k = 0 \\
$ or $
\left( {k - 6} \right) = 0 \\
\Rightarrow k = 6 \\
$
But here k=0 is neglected because if k=0 the given quadratic equation $kx\left( {x - 2} \right) + 6 = 0$ reduces to 6=0 which is not true.
So, k=6 is the only value for which the given quadratic equation has two equal roots.
Note: In these types of problems, the values of the roots corresponding to the any quadratic equation \[a{x^2} + bx + c = 0\] are given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$. If the quadratic equation has two equal roots then the value of that root is given by $x = \dfrac{{ - b}}{{2a}}$.
Complete step-by-step answer:
The given quadratic equation is \[
kx\left( {x - 2} \right) + 6 = 0 \\
\Rightarrow k{x^2} - 2kx + 6 = 0{\text{ }} \to {\text{(1)}} \\
\]
As we know that for any general quadratic equation \[a{x^2} + bx + c = 0{\text{ }} \to {\text{(2)}}\] to have two equal roots, the discriminant should be equal to zero where the discriminant is given by
${\text{d}} = \sqrt {{b^2} - 4ac} \to {\text{(3)}}$
By comparing equations (1) and (2), we get
a=k, b=-2k and c=6
It is also given that the given quadratic equation given by equation (1) has two equal roots.
So, d=0 for the given quadratic equation.
$
\Rightarrow \sqrt {{{\left( { - 2k} \right)}^2} - 4\left( k \right)\left( 6 \right)} = 0 \\
\Rightarrow \sqrt {4{k^2} - 24k} = 0 \\
\Rightarrow 4{k^2} - 24k = 0 \\
\Rightarrow 4k\left( {k - 6} \right) = 0 \\
$
Either $
4k = 0 \\
\Rightarrow k = 0 \\
$ or $
\left( {k - 6} \right) = 0 \\
\Rightarrow k = 6 \\
$
But here k=0 is neglected because if k=0 the given quadratic equation $kx\left( {x - 2} \right) + 6 = 0$ reduces to 6=0 which is not true.
So, k=6 is the only value for which the given quadratic equation has two equal roots.
Note: In these types of problems, the values of the roots corresponding to the any quadratic equation \[a{x^2} + bx + c = 0\] are given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$. If the quadratic equation has two equal roots then the value of that root is given by $x = \dfrac{{ - b}}{{2a}}$.
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
Which are the Top 10 Largest Countries of the World?
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE