Question

10.) Find the values of p for which the quadratic equation

$(p + 1)x {}^{2} - 6(p + 1)x + 3(p + 9) = 0$

, p # - 1 has equal roots. Hence, find the roots of the equation.

Answer 1

Given Equation is (p + 1)x^2 - 6(p + 1)x + 3(p + 9) = 0.

Here, a = (p + 1), b = -6(p + 1), c = 3(p + 9).

Given that the equation has equal roots.

∴ D = b^2 - 4ac = 0

      = [-6(p + 1)]^2 - 4(p + 1)(3p + 27) = 0

      = 36(p^2 + 1 + 2p) - 4(3p^2 + 27p + 3p + 27) = 0

      = 36p^2 + 36 + 72p - 12p^2 - 108p - 12p - 108p = 0

      = 36p^2 + 72p + 36 - 12p^2 - 120p - 108 = 0

      = 24p^2 - 48p - 72 = 0

      = 2p^2 - 4p - 6 = 0

      = p^2 - 2p - 3 = 0

      = p^2 + p - 3p - 3 = 0

      = p(p + 1) - 3(p + 1) = 0

      = (p - 3)(p + 1) = 0

      = p = 3,-1.

Therefore, the value of p = 3,-1.

Hope it helps!

Answer 2

SOLUTION :  

Given : (p + 1)x² - 6(p + 1)x + 3(p + 9) = 0, p ≠ -1 has equal roots ………(1)

On comparing the given equation with ax² + bx + c = 0  

Here, a = p + 1 , b = - 6(p +1)  , c = 3(p +9)

D(discriminant) = b² – 4ac

D = [- 6(p +1)² - 4 × (p + 1) × 3(p + 9)

D = [36((p)² + 1²+ 2× p× 1)) - 12(p²  + 9p + p + 9)

[(a + b)² = a² + b² + 2ab]

D = 36(p² + 1 + 2p - 12(p²  + 10p + 9)

D = 36p² + 36 + 72p - 12 p² - 120 p - 108

D = 36p² - 12 p² + 72p - 120p + 36 - 108  

D = 24p² - 48p - 72

Given :  Equal roots  

Therefore , D = 0

24p² - 48p - 72  = 0

24(p² - 2p - 3) = 0  

p² - 2p - 3 = 0  

p² - 3p + p - 3 = 0  

[By middle term splitting]

p(p - 3) + 1 (p - 3) = 0

(p + 1) (p - 3) = 0

p + 1  = 0  or (p - 3) = 0

p = - 1  or p = 3

The value of p is - 1  & 3 .

It is given that p ≠ - 1 , so p = 3

Hence , the value of p is 3 only.

On putting p = 3 in eq 1 ,

(p + 1)x² - 6(p + 1)x + 3(p + 9) = 0

(3 + 1)x² - 6(3 + 1)x + 3(3 + 9) = 0

4x² - 6(4)x + 3(12) = 0

4x² - 24x + 36 = 0

4(x² - 6x + 9) = 0

x² - 6x + 9 = 0

x² - 3x  - 3x + 9 = 0

[By middle term splitting]

x(x - 3) -3(x - 3) = 0

(x - 3) = 0  or  (x - 3) = 0

x = 3  or  x = 3  

Roots are 3 & 3

Hence ,the roots of the equation (p + 1)x² - 6(p + 1)x + 3(p + 9) = 0 is 3 .

★★ NATURE OF THE ROOTS

If D = 0 roots are real and equal  

If D > 0 roots are real and distinct

If D < 0  No real roots  

HOPE THIS ANSWER WILL HELP YOU…