Answer

Verified

449.1k+ 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

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

How many crores make 10 million class 7 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths