[tex]\sf\implies k = 6 \: \: \: or \: \: \: k = 0 \: \{rejected \} \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Concept Used :-
Nature of roots :-
Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.
Now, three cases arises :
If Discriminant, D > 0, then roots of the equation are real and unequal.
If Discriminant, D = 0, then roots of the equation are real and equal.
If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.
[tex]\:\:\:\:\:\:\:\:\:\:[/tex] [tex]\implies\sf\bf{ k = 6}[/tex]
[tex]\rule{200pt}{4pt}[/tex]
Concept Used :-
Nature of roots :-
Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.
Now, three cases arises :
If Discriminant, D > 0, then roots of the equation are real and unequal.
If Discriminant, D = 0, then roots of the equation are real and equal.
If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.
Answers & Comments
Verified answer
Answer:
[tex] \qquad \:\boxed{ \bf{ \:k \: = \: 6 \: }} \\ \\ [/tex]
Step-by-step explanation:
Given that the quadratic equation kx(x - 2) + 6 = 0 have two equal roots.
The above equation can be rewritten as
[tex]\sf \: {kx}^{2} - 2kx + 6 = 0 \\ \\ [/tex]
On comparing with ax² + bx + c = 0, we get
[tex] \qquad \:\boxed{\begin{aligned}&\sf \: a=k\\ \\ &\sf \: b= - 2k \\ \\&\sf \: c=6\end{aligned}} \\ \\ [/tex]
Now, it is given that, equation has equal roots.
[tex]\sf \: Discriminant = 0 \\ \\ [/tex]
[tex]\sf \: {b}^{2} - 4ac = 0 \\ \\ [/tex]
[tex]\sf \: {( - 2k)}^{2} - 4(k)(6) = 0 \\ \\ [/tex]
[tex]\sf \: {4k}^{2} - 24k = 0 \\ \\ [/tex]
[tex]\sf \: 4k(k - 6) = 0 \\ \\ [/tex]
[tex]\sf\implies k = 6 \: \: \: or \: \: \: k = 0 \: \{rejected \} \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Concept Used :-
Nature of roots :-
Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.
Now, three cases arises :
Where,
Discriminant, D = b² - 4ac
[tex]\underline{\underline{\bf{Question: -}}}[/tex]
Find the value of k so that the Quadratic Equations [tex]\sf{Kx (x - 2) + 6 = 0}[/tex] has two equal roots.
[tex]\underline{\underline{\bf{Answer : -}}}[/tex]
[tex]\:\:\:\:\:\:\:\:\:\:[/tex] [tex]\implies\sf{Kx (x - 2) + 6 = 0}[/tex]
[tex]\:\:\:\:\:\:\:\:\:\:[/tex] [tex]\implies\sf\bf{D = b² - 4ac = 0}[/tex]
[tex]\:\:\:\:\:\:\:\:\:\:[/tex] [tex]\implies\sf{D = (-2k)² - 4 (k) (6) = 0}[/tex]
[tex]\:\:\:\:\:\:\:\:\:\:[/tex] [tex]\implies\sf{4k² - 24k = 0}[/tex]
[tex]\:\:\:\:\:\:\:\:\:\:[/tex] [tex]\implies\sf{k - 6 = 0}[/tex]
[tex]\:\:\:\:\:\:\:\:\:\:[/tex] [tex]\implies\sf\bf{ k = 6}[/tex]
[tex]\rule{200pt}{4pt}[/tex]
Concept Used :-
Nature of roots :-
Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.
Now, three cases arises :
If Discriminant, D > 0, then roots of the equation are real and unequal.
If Discriminant, D = 0, then roots of the equation are real and equal.
If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.