Question

Solve number 5
Answer :-
1). -1,2/3
2). 8

Answer 1

Solution:-

Since, It has equal Roots and the Equations are equal to zero.

=) Discriminant (D) = 0

(1). x² + 4kx + ( k² - k +2) = 0

=) D = b² - 4ac

=) 0 = (4k)² - 4 × 1 × ( k² - k +2)

=) 16k² - 4k² + 4k - 8 = 0

=) 4 ( 4k² - k² + k - 2) = 0

=) 4k² - k² + k - 2 = 0

=) 3k² + k - 2 = 0

=) 3k² + ( 3 - 2)k -2 = 0

=) 3k² + 3k - 2k -2=0

=) 3k ( k + 1) -2 ( k +1) = 0

=) ( k +1) ( 3k -2)

=) k = -1. and. k = 2/3.

(2). ( k -4)x² + 2(k -4)x + 4 = 0

=) D = 0

=) b² - 4ac = 0

=) [ 2(k-4)]² - 4 × ( k-4) × 4 = 0

=) [ 4( k -4)²] - 4 × 4 × ( k-4) = 0

=) 4 [ ( k-4)² - 4( k-4)] = 0

=) ( k-4)² - 4( k-4) = 0

=) k² + 16 - 8k - 4k + 16 = 0

=) k² - 12k + 32 = 0

=) k² - ( 8 + 4)k + 32 = 0

=) k² - 8k - 4k + 32= 0

=) k ( k -8) -4 ( k -8) = 0

=) ( k -8). ( k-4)

=) k = 8. and. k= 4.

Hence Solved!!

Identify Used:-

( a - b)² = a² + b² - 2ab.

Answer 2

SOLUTION_✌

GIVEN

• $x^{2} + 4kx + (k^{2}  - k +2) =0$ has equal roots.

⇒Hence, $b^{2}  - 4ac$

⇒That is, $4[tex]4k^{2} -4(k^{2}-k+2)=0\\4k^{2} -4k^{2} +4k -8=0\\16k^{2} -4k^{2} +4k -8=0\\4(4k^{2} -k^{2} +k-2)=0\\4k^{2}  - k^{2} +k-2=0\\3k^{2} +k -2=0\\3k^{2} +(3-2)k-2=0\\3k^{2} +3k-2k-2=0\\3k(k+1) -2(k+1)=0\\(k+1)=0 or (3k-2)=0\\k= -1 or 3k=2\\k= -1 and k= \frac{2}{3}$k^{2}  - 4.1.(k^{2} -k+2)[/tex] =0