Find the value(s) of k for which each of the following quadratic equation has equal roots: (i) 3kx 2 = 4(kx - 1) (ii) (k + 4)x 2 + (k + 1)x + 1 = 0 Also, find the roots for that value(s) of k in each case.

(i) Given, Comparing it with ax 2 + bx + c = 0 a= 3k, b = -4k, c = 4 Given, Equation has equal roots ∴ b 2 - 4ac = 0 But k cannot be equal to 0 as that will make a = 3k = 0 , which will make roots equal to infinity. ∴ k = 3 Hence equation is 9x 2 - 12x + 4 = 0 Hence the value of k is 3 and the roots are (ii) The given equation is (k + 4)x 2 +(k + 1)x + 1 = 0 Comparing it with ax 2 + bx + c = 0 a = (k + 4), b = (k + 1), c = 1 Given, Equation has equal roots ∴ b 2 - 4ac = 0 ∴ When k = -3 , equation is x 2 - 2x + 1 = 0 ∴ When k = 5 , equation is 9x 2 + 6x + 1 = 0 k = -3, 5 When k = -3 , roots are 1, 1 When k = 5 , roots are

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Mathematics

Find the value(s) of k for which each of the following quadratic equation has equal roots:
(i) 3kx² = 4(kx - 1)
(ii) (k + 4)x² + (k + 1)x + 1 = 0
Also, find the roots for that value(s) of k in each case.

Quadratic Equations

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Mathematics

Find the value(s) of k for which each of the following quadratic equation has equal roots:
(i) 3kx² = 4(kx - 1)
(ii) (k + 4)x² + (k + 1)x + 1 = 0
Also, find the roots for that value(s) of k in each case.

Quadratic Equations

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Mathematics

Find the value(s) of k for which each of the following quadratic equation has equal roots:(i) 3kx2 = 4(kx - 1)(ii) (k + 4)x2 + (k + 1)x + 1 = 0Also, find the roots for that value(s) of k in each case.

Quadratic Equations

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Find the values of k so that the quadratic equation (4 - k)x2 + 2(k + 2)x + (8k + 1) = 0 has equal roots.

Find the values of m so that the quadratic equation 3x2 - 5x - 2m = 0 has two distinct real roots.

Find two natural numbers which differ by 3 and whose squares have the sum 117.

Divide 16 into two parts such that twice the square of the larger part exceeds the square of the smaller part by 164.

Find the value(s) of k for which each of the following quadratic equation has equal roots:
(i) 3kx² = 4(kx - 1)
(ii) (k + 4)x² + (k + 1)x + 1 = 0
Also, find the roots for that value(s) of k in each case.

Find the values of k so that the quadratic equation (4 - k)x² + 2(k + 2)x + (8k + 1) = 0 has equal roots.

Find the values of m so that the quadratic equation 3x² - 5x - 2m = 0 has two distinct real roots.