Question

find the values of k for which the quadratic equation 2xsquare-kx+k=0 has equal roots​

Answer 1

Question:

Find the value of k for which the quadratic equation 2x² -kx + k= 0 has equal roots.

Answer:

k = 0 or 8

Note:

• An equation of degree 2 is know as quadratic equation .

• Roots of an equation is defined as the possible values of the unknown (variable) for which the equation is satisfied.

• The maximum number of roots of an equation will be equal to its degree.

• A quadratic equation has atmost two roots.

• The general form of a quadratic equation is given as , ax² + bx + c = 0 .

• A quadratic equation has atmost two roots.

• The general form of a quadratic equation is given as , ax² + bx + c = 0 .

• The discriminant of the quadratic equation is given as , D = b² - 4ac .

• If D = 0 , then the quadratic equation would have real and equal roots .

• If D > 0 , then the quadratic equation would have real and distinct roots .

• If D < 0 , then the quadratic equation would have imaginary roots .

Solution:

The given quadratic equation is ;

2x² - kx + k = 0 .

Clearly , we have ;

a = 2

b = -k

c = k

We know that ,

The quadratic equation will have real and equal roots if its discriminant is zero .

=> D = 0

=> (-k)² - 4•2•k = 0

=> k² - 8k = 0

=> k•(k-8) = 0

=> k = 0 or 8

Hence,

The required values of k are 0 and 8 .

Answer 2

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Value\:of\:k=0\:and\:8}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{ \underline \bold{Given : }} \\ \tt{ : \implies 2x^{2} -kx + k = 0 }\\ \\ \red{ \underline \bold{To \: Find : }} \\ \tt{: \implies value \: of \: k = ?}$

• According to given question :

$\tt{ : \implies 2x^{2} -kx + k= 0} \\ \\ \tt{\circ \: a = 2} \\ \\ \tt{\circ \: b = -k}\\\\ \tt{\circ \:c = k}\\ \\ \bold{Discriminant \: = 0} \\ \\ \tt{: \rightarrow \: D \implies {b}^{2} - 4ac = 0 } \\ \\ \tt{: \implies {b}^{2} - 4ac = 0} \\ \\ \text{Putting \: the \: given \: values} \\ \tt{: \implies (-k)^{2} - 4\times2 \times k= 0 } \\ \\ \tt{: \implies \: {k}^{2} -8k = 0 } \\ \\ \tt{ : \implies \: k({k} - 8) = 0 } \\\\ \tt{: \implies k= 0} \\ \\ \tt{: \implies k= 8} \\ \\ \green{\tt{: \implies k = 0\:and\:8}}$