Sure, let's solve this problem step-by-step.
For a quadratic equation to have equal roots, its discriminant must be zero. The given quadratic equation is:
[tex]\[ k x^2 + 2 k (1 - x) + 1 - 4 x = 0 \][/tex]
First, let's simplify the equation. Expanding the terms inside the parentheses, we have:
[tex]\[ k x^2 + 2 k - 2 k x + 1 - 4 x = 0 \][/tex]
Next, we group the like terms together:
[tex]\[ k x^2 - 2 k x - 4 x + 2 k + 1 = 0 \][/tex]
Now, let's rewrite this in the standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[ k x^2 + (-2 k - 4) x + (2 k + 1) = 0 \][/tex]
Here, the coefficients are:
- [tex]\( a = k \)[/tex]
- [tex]\( b = -2 k - 4 \)[/tex]
- [tex]\( c = 2 k + 1 \)[/tex]
For the quadratic equation to have equal roots, its discriminant (Δ) must be zero. The discriminant of a quadratic equation [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4 a c \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[ \Delta = (-2 k - 4)^2 - 4 (k) (2 k + 1) \][/tex]
Next, we set the discriminant to zero because we need equal roots:
[tex]\[ (-2 k - 4)^2 - 4 k (2 k + 1) = 0 \][/tex]
Now, expand and simplify the expression:
[tex]\[ (4 k^2 + 16 k + 16) - 4 k (2 k + 1) = 0 \][/tex]
[tex]\[ 4 k^2 + 16 k + 16 - 8 k^2 - 4 k = 0 \][/tex]
Combine the like terms:
[tex]\[ -4 k^2 + 12 k + 16 = 0 \][/tex]
Divide everything by -4 for simplicity:
[tex]\[ k^2 - 3 k - 4 = 0 \][/tex]
Now, solve this quadratic equation for [tex]\(k\)[/tex]. We factorize it:
[tex]\[ (k - 4)(k + 1) = 0 \][/tex]
Setting each factor to zero gives us the solutions:
[tex]\[ k - 4 = 0 \quad \text{or} \quad k + 1 = 0 \][/tex]
Thus,
[tex]\[ k = 4 \quad \text{or} \quad k = -1 \][/tex]
Therefore, the values of [tex]\(k\)[/tex] for which the equation [tex]\( k x^2 + 2 k (1 - x) + 1 - 4 x = 0 \)[/tex] will have equal roots are:
[tex]\[ \boxed{k = -1 \ \text{or} \ k = 4} \][/tex]
