Question

Find value of k for which quadratic equation 9x²-3x+k has equal roots

Answer 1

Question:

Find the value of k for which the quadratic equation 9x² - 3x + k = 0 has equal roots.

Answer:

k = 1/4

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 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 ;

9x² - 3x + k = 0

Clearly , we have ;

a = 9

b = -3

c = k

We know that ,

The quadratic equation will have equall roots if its discriminant is equal to zero .

=> D = 0

=> (-3)² - 4•9•k = 0

=> 9 - 9•4•k = 0

=> 9•(1 - 4k) = 0

=> 1 - 4k = 0

=> 4k = 1

=> k = 1/4

Hence,

The required values of k is 1/4 .

Answer 2

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

$\green{\tt{\therefore{Value\:of\:k=\frac{1}{4}}}}$

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

$\green{ \underline \bold{Given : }} \\ \tt{ : \implies 9x^{2} -3x + k = 0 }\\ \\ \red{ \underline \bold{To \: Find : }} \\ \tt{: \implies value \: of \: k = ?}$

• According to given question :

$\tt{ : \implies 9x^{2} +3x + k= 0} \\ \\ \tt{\circ \: a = 9} \\ \\ \tt{\circ \: b = -3}\\\\ \tt{\circ \:c = k}\\ \\ \bold{Discriminant \: = 0} \\ \\ \tt{: \rightarrow \: D \implies {b}^{2} - 4ac = 0 } \\ \\ \tt{: \implies {b}^{2} - 4ac = 0} \\ \\ \text{Putting \: the \: given \: values} \\ \tt{: \implies( -3)^{2} - 4\times9\times k= 0 } \\ \\ \tt{: \implies \: 9 -36k = 0 } \\ \\ \tt{ : \implies \: 9(1 - 4k) = 0 } \\\\ \tt{: \implies 1-4k=0} \\ \\ \tt{: \implies 4k= 1} \\ \\ \green{\tt{: \implies k = \frac{1}{4} }}$