Question

find the values of k for which the quadratic equation x^2+8x+2k has no real roots​

Answer 1

Question:

Find the value of k for which the quadratic equation x² + 8x + 2k = 0 has no real roots.

Answer:

k € (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 .

• 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 ;

x² + 8x + 2k = 0

Clearly , we have ;

a = 1

b = 8

c = 2k

We know that ,

The quadratic equation will have non-real(imaginary) roots if its discriminant is less than zero .

=> D < 0

=> 8² - 4•1•2k < 0

=> 64 - 8k < 0

=> 64 < 8k

=> 8k > 64

=> k > 64/8

=> k > 8

=> k € (8,∞)

Hence,

The required values of k are (8,∞)

Answer 2

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

$\green{\tt{\therefore{Value\:of\:k\:\in\:(8,\infty)}}}$

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

$\green{ \underline \bold{Given : }} \\ \tt{ : \implies x^{2} +8x +2k = 0 }\\ \\ \red{ \underline \bold{To \: Find : }} \\ \tt{: \implies value \: of \: k = ?}$

• According to given question :

$\tt{ : \implies x^{2} +8x + 2k= 0} \\ \\ \tt{\circ \: a = 1} \\ \\ \tt{\circ \: b = 8}\\\\ \tt{\circ \:c = 2k}\\ \\ \bold{Discriminant \: <0} \\ \\ \tt{: \rightarrow \: D \implies {b}^{2} - 4ac < 0 } \\ \\ \tt{: \implies {b}^{2} - 4ac =<0} \\ \\ \text{Putting \: the \: given \: values} \\ \tt{: \implies (8)^{2} - 4\times1 \times 2k<0 } \\ \\ \tt{: \implies \: 64 -8k<0 } \\ \\ \tt{ : \implies \: 64<8k} \\\\ \tt{: \implies 8k>64} \\ \\ \tt{: \implies k> \frac{\cancel{64}}{\cancel{8}}} \\ \\ \green{\tt{: \implies k > 8 }}\\ \\ \tt{\green{\therefore Value\:of\:k\:\in\:(8,\infty)}}$