Question

If kx^2-5x+3=0 and 2x^2-kx+1=0 have equal discriminants, then find the value of k.

Answer 1

Answer:

ax^2 +bx+ c =0 and Discriminant for this equation is b^2- 4ac. Since it is given that

kx^2-5x+3 =0 and 2x^2-kx+1=0 have equal discriminants, 25- 4(3)k = k^2 -4(2)(1) =>

k^2+12k-33 =0 has two roots

$\frac{ - 12 + \sqrt{ {12}^{2} - 4( - 33) } }{2}$

and

$\frac{ - 12 - \sqrt{ {12}^{2} - 4( - 33)} }{2}$

can be simplified as

$\frac{ - 12 + \sqrt{276} }{2} \: and \: \frac{ - 12 - \sqrt{276} }{2}$

So the values of k are -6-√69 and -6+√69. Hope it helps you. Mark me as Brainliest if you think I am worth of it.

Answer 2

The values of k are  - 6 + √69  and   - 6 - √69.

Given: The equations kx² - 5x + 3 = 0 and 2x² - kx + 1 = 0 have equal discriminants.

To Find: The value of k.

Solution:

We know that the determinant of a given polynomial ax² + bx + c = 0, can be found by applying the formula,

               D = b² - 4ac                                                              ......(1)

Where D = determinant, a,b,c = constants of the given polynomial.

Coming to the numerical, we are given;

The equation: kx² - 5x + 3 = 0                                                 .......(2)

Here, a = k, b = - 5, c = 3.

The determinant of equation (2), can be found by putting the respective values in (1),

                    D = b² - 4ac    

                ⇒  D = ( - 5 )² - 4 × k × 3

                ⇒  D = 25 - 12k                                                        .........(3)

We are also given;

The equation: 2x² - kx + 1 = 0                                                 .......(4)

Here, a = 2, b = - k, c = 1.

The determinant of equation (2), can be found by putting the respective values in (1),

                     D = b² - 4ac    

                ⇒  D = ( - k )² - 4 × 2 × 1

                ⇒  D = k² - 8                                                        .........(5)

Now, as it is said that the determinants are equal, so we can equate (3
) and (4) to get;

                    25 - 12k = k² - 8  

                ⇒ k² + 12k - 33 = 0

                ⇒ k = [ - b ± √( b² - 4ac )] / 2a

                ⇒ k =  [ - 12 ± √( 12² - 4 × 1 × ( - 33 ))] / 2

                ⇒ k =  [ - 12 ± √276 ] / 2

                ⇒ k = [ - 12 ± 2√69 ] / 2

                ⇒ k = - 6 ± √69

So, the values of k are  - 6 + √69  and   - 6 - √69.

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