Question

for which value of the quadratic polynomial 4 x square -1 2 x minus K is equal to zero has no real roots​

Answer 1

Question :--- for which value of the quadratic polynomial 4x² -1 2x -K =0 has no real roots.....

Concept used :---

If A•x^2 + B•x + C = 0 ,is any quadratic equation,

then its discriminant is given by;

D = B^2 - 4•A•C

• If D = 0 , then the given quadratic equation has real and equal roots.

• If D > 0 , then the given quadratic equation has real and distinct roots.

• If D < 0 , then the given quadratic equation has unreal (imaginary) roots...

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Solution :----

From 4x² -1 2x -K = 0 we have ,

→ a = 4

→ b = (-12)

→ c = (-k)

Since , now real roots, than D < 0.

Putting values we get,

→ (-12)² - 4*4*(-k) < 0

→ 144 + 16k < 0

→ 16k < (-144)

Dividing both sides by 16

→ k < (-9) .

Hence, value of k will be less than (-9) for no real roots.

k = { (-10) to - infinity } .

Answer 2

$\huge\star{\green{\underline{\mathfrak{Answer :-}}}}$

Quadratic polynomial :- $\tt {4 {x}^{2}-12x-k=0}$

If it has no real roots, then the $\bold {\tt D < 0}$

$\tt {\therefore {b}^{2}-4ac <0}$

In the equation,

$\tt{ a = 4}$

$\tt {b = (-12)}$

$\tt{c = (-k)}$

Putting in the values,

$\leadsto \tt { {(-12)}^{2}- 4 (4)(-k)<0}$

$\leadsto \tt {144+16k <0}$

$\leadsto \tt {16k <-144}$

$\leadsto \tt {k < \dfrac {-144}{16}}$

$\huge\leadsto\tt{\boxed {k < (-9)}}$

So, the value of k will be from (-10) to infinity.

$\rule{200}2$

The quadratic equation $\tt{a {x}^{2}+bx+c=0}$ will have roots that are :-

• If discriminant is greater than zero,then the roots will be real and distinct

• If discriminant is less than zero, then the roots will be non real.

• If discriminant is equal to zero then the roots will be equal and real.

$\rule{200}2$