Find the nature of roots of the following quadratic equations : (i) (ii) If real roots exist, find them.

(i) The given equation is Comparing it with ax 2 + bx + c = 0, we get a = 1 , b = , c = ∴ Discriminant = b 2 - 4ac Putting values of a, b, c in formula Discriminant = Since Discriminant 0, hence the given equation has two distinct real roots. The roots of the equation are given by: Hence roots of the given equations are 1 , . (ii) The given equation is Comparing it with ax 2 + bx + c = 0, we get a = 1 , b = , c = -1 ∴ Discriminant = b 2 - 4ac Putting values of a, b, c in formula Discriminant = 16 , Since Discriminant 0, hence given equation have real and distinct roots. The roots of the equation are given by Hence roots of the given equations are , .

Find the discriminant of the following quadratic equations and hence find the nature of roots :
(i) 3x² - 5x - 2 = 0
(ii) 2x² - 3x + 5 = 0
(iii) 16x² - 40x + 25 = 0
(iv) 2x² + 15x + 30 = 0

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Without solving the following quadratic equations, find the value of 'p' for which the given equations have real and equal roots :
(i) px² - 4x + 3 = 0
(ii) x² + (p - 3)x + p = 0

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Find the values of k for which each of the following quadratic equation has equal roots :
(i) x² + 4kx + (k² - k + 2) = 0
(ii) (k - 4)x² + 2(k - 4)x + 4 = 0

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Mathematics

Find the nature of roots of the following quadratic equations : (i) (ii) If real roots exist, find them.

Quadratic Equations

Related Questions

Find the discriminant of the following quadratic equations and hence find the nature of roots :
(i) 3x² - 5x - 2 = 0
(ii) 2x² - 3x + 5 = 0
(iii) 16x² - 40x + 25 = 0
(iv) 2x² + 15x + 30 = 0

Without solving the following quadratic equations, find the value of 'p' for which the given equations have real and equal roots :
(i) px² - 4x + 3 = 0
(ii) x² + (p - 3)x + p = 0

Find the values of k for which each of the following quadratic equation has equal roots :
(i) x² + 4kx + (k² - k + 2) = 0
(ii) (k - 4)x² + 2(k - 4)x + 4 = 0

Mathematics

Find the nature of roots of the following quadratic equations : (i) (ii) If real roots exist, find them.

Quadratic Equations

Related Questions

Find the discriminant of the following quadratic equations and hence find the nature of roots :(i) 3x2 - 5x - 2 = 0(ii) 2x2 - 3x + 5 = 0(iii) 16x2 - 40x + 25 = 0(iv) 2x2 + 15x + 30 = 0

Without solving the following quadratic equations, find the value of 'p' for which the given equations have real and equal roots :(i) px2 - 4x + 3 = 0(ii) x2 + (p - 3)x + p = 0

Find the values of k for which each of the following quadratic equation has equal roots :(i) x2 + 4kx + (k2 - k + 2) = 0(ii) (k - 4)x2 + 2(k - 4)x + 4 = 0

Find the discriminant of the following quadratic equations and hence find the nature of roots :
(i) 3x² - 5x - 2 = 0
(ii) 2x² - 3x + 5 = 0
(iii) 16x² - 40x + 25 = 0
(iv) 2x² + 15x + 30 = 0

Without solving the following quadratic equations, find the value of 'p' for which the given equations have real and equal roots :
(i) px² - 4x + 3 = 0
(ii) x² + (p - 3)x + p = 0

Find the values of k for which each of the following quadratic equation has equal roots :
(i) x² + 4kx + (k² - k + 2) = 0
(ii) (k - 4)x² + 2(k - 4)x + 4 = 0