Question

find the values of k for the quadratic equation kx(x-2)+6=0,so that it has two equal roots

Answer 1

the given equation can be written in the form of
$k {x}^{2} - 2kx + 6 = 0$
now it has equal roots that means The Discriminant is 0
D = 0
Also
${b}^{2} - 4ac = 0 \\ {4k}^{2} - 4(k)(6) = 0 \\ 4 {k}^{2} - 24k = 0 \\ 4k(k - 6) = 0 \\ k - 6 = 0 \\ k = 6$
Hence the value of K is 6.

Answer 2

The value of k = 6

Given :

The quadratic equation kx(x - 2) + 6 = 0

To find :

The value of k so that the quadratic equation kx(x - 2) + 6 = 0 has two equal roots

Concept :

1. General form of a quadratic equation is

ax² + bx + c = 0

The Discriminant of the quadratic equation is denoted by D and defined as

D = b² - 4ac

2. If a quadratic equation has equal roots then discriminant of the equation = 0

Solution :

Step 1 of 3 :

Write down the given Quadratic equation

Here the given Quadratic equation is

kx(x - 2) + 6 = 0

Step 2 of 3 :

Find the discriminant

kx(x - 2) + 6 = 0

⇒ kx² - 2kx + 6 = 0

Comparing with the general equation ax² + bx + c = 0 we get

a = k , b = - 2k , c = 6

Hence discriminant of the quadratic equation

= b² - 4ac

$\sf = {( -2k) }^{2} - 4 \times k \times 6$

$\sf = 4{k} ^{2} - 24k$

$\displaystyle \sf{ =4k(k - 6) }$

Step 3 of 3 :

Find the value of k

Since the quadratic equation has equal roots

∴ Discriminant of the equation = 0

⇒ 4k(k - 6) = 0

⇒ k = 0 , 6

When k = 0

The quadratic equation becomes 6 = 0

Which is absurd

Hence the required value of k = 6

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