Question

find the value of k for which roots of the equation
$x {}^{2} - 8kx + 2k = 0$
are equal
​

Answer 1

Question:

Find he value(values) of k for which the quadrtic equation x² - 8kx + 2k has has equal roots.

Answer:

k = 0 , 1/8 .

Note:

• The possible values of x which satisfy the equation(in variable x ) are called its roots .

• A quadratic equation can have at most two roots.

• The discriminantnant D of a quadratic equation ,

ax² + bx + c = 0 , is given by;

D = b² - 4ac .

• If D = 0 , then both the roots of the quadratic equation are real and equal.

• If D > 0 , then both the roots of the quadratic equation are real and distinct.

• If D < 0 , then both the roots of the quadratic equation are imaginary.

Solution:

The given quadratic equation is;

x² - 8kx + 2k = 0.

Also,

The discriminant of the given quadratic equation will be;

=> D = (-8k)² - 4•1•2k

=> D = 64k² - 8k

=> D = 8k(8k - 1)

It is given that ,

The given quadratic equation has equal roots , hence its discriminant must be zero.

ie ;

=> D = 0

=> 8k(8k - 1) = 0

=> k(8k - 1) = 0

=> k = 0 OR (8k - 1) = 0

=> k = 0 OR 8k = 1

=> k = 0 OR k = 1/8

Hence,

The required values of k are :

0 and 1/8 .

Moreover;

Case1 : When k = 0

Then , the given quadratic equation will reduce to ;

=> x² - 8kx + 2k = 0

=> x² - 8•0•x + 2•0 = 0

=> x² = 0

Also,

Roots of the quadratic equation will be ;

x = 0 , 0

Case2 : When k = 4

Then , the given quadratic equation will reduce to ;

=> x² - 8kx + 2k = 0

=> x² - 8•(1/8)•x + 2•(1/8) = 0

=> x² - x + 1/4 = 0

=> x² - 2•x•(1/2) + (1/2)² = 0

=> (x - 1/2)² = 0

Also,

Roots of the quadratic equation will be ;

x = 1/2 , 1/2

Hence,

If k = 0 , then the given quadratic equation will reduce to x² = 0 and its equal roots will be 0 , 0 .

If k = 1/8 , then the given quadratic equation will reduce to (x - 1/2)² = 0 and its equal zeros will be 1/2 , 1/2 .

Answer 2

Answer:

$\huge{\underline{\underline{\mathfrak{Answer\::}}}}$

K = 0 or K = 1/8

________________________

The given quadratic equation :-

${x}^{2}-8kx+2k=0$

Roots of the equation are equal.

Using the standard equation , $a{x}^{2}+bx+c$ we get,

a = 1 , b = -8k and c = 2k

And the discriminant is :

${b}^{2}-4ac$

=> Discriminant={-8k}^2 -4 × 1 × 2k

=> Discriminant = 64k^2 - 8k

=> Discriminant = 8k(8k- 1)

As the equation has equal roots so here the discriminant is obviously 0.

=> 8k ( 8k - 1) = 0

=> 8k = 0 or 8k-1 = 0

=> K = 0 or 8k = 1

=> K = 0 or K = 1/8

Hope it helps you ♥ ♥ ♥