To determine the value of [tex]\( k \)[/tex] for which the quadratic equation [tex]\((k + 1)x^2 - 6(k + 1)x + 3(k + 9) = 0\)[/tex] has real and equal roots, we need to utilize the discriminant condition for quadratic equations.
The general quadratic equation is given by:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
The discriminant for this equation is [tex]\( \Delta = b^2 - 4ac \)[/tex].
For the equation to have real and equal roots, the discriminant must be zero:
[tex]\[ \Delta = 0 \][/tex]
In the given quadratic equation:
[tex]\[ a = k + 1 \][/tex]
[tex]\[ b = -6(k + 1) \][/tex]
[tex]\[ c = 3(k + 9) \][/tex]
Substitute these coefficients into the discriminant formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = [-6(k + 1)]^2 - 4(k + 1)[3(k + 9)] \][/tex]
Simplify the expression:
[tex]\[ \Delta = 36(k + 1)^2 - 4(k + 1) \cdot 3(k + 9) \][/tex]
[tex]\[ \Delta = 36(k + 1)^2 - 12(k + 1)(k + 9) \][/tex]
Expand both terms:
[tex]\[ 36(k + 1)^2 = 36(k^2 + 2k + 1) = 36k^2 + 72k + 36 \][/tex]
[tex]\[ 12(k + 1)(k + 9) = 12(k^2 + 10k + 9) = 12k^2 + 120k + 108 \][/tex]
Combine these results into the discriminant expression:
[tex]\[ \Delta = 36k^2 + 72k + 36 - (12k^2 + 120k + 108) \][/tex]
Simplify by combining like terms:
[tex]\[ \Delta = 36k^2 + 72k + 36 - 12k^2 - 120k - 108 \][/tex]
[tex]\[ \Delta = 24k^2 - 48k - 72 \][/tex]
Set the discriminant to zero for real and equal roots:
[tex]\[ 24k^2 - 48k - 72 = 0 \][/tex]
Divide the entire equation by 24 to simplify:
[tex]\[ k^2 - 2k - 3 = 0 \][/tex]
Now, solve the quadratic equation [tex]\( k^2 - 2k - 3 = 0 \)[/tex] using factorization:
[tex]\[ k^2 - 2k - 3 = (k - 3)(k + 1) = 0 \][/tex]
Setting each factor to zero gives us the values of [tex]\( k \)[/tex]:
[tex]\[ k - 3 = 0 \quad \Rightarrow \quad k = 3 \][/tex]
[tex]\[ k + 1 = 0 \quad \Rightarrow \quad k = -1 \][/tex]
Since [tex]\( k \neq -1 \)[/tex] is given in the problem, we discard the solution [tex]\( k = -1 \)[/tex].
Thus, the value of [tex]\( k \)[/tex] for which the quadratic equation has real and equal roots is:
[tex]\[ k = 3 \][/tex]
