To determine the value of [tex]\(\lambda\)[/tex] for which the quadratic equation [tex]\(\lambda x^2 - 4x + 4 = 0\)[/tex] has equal roots, we need to use the condition for equal roots in a quadratic equation. The condition is that the discriminant must be zero.
The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
In our equation [tex]\(\lambda x^2 - 4x + 4 = 0\)[/tex], the coefficients are:
[tex]\[
a = \lambda, \quad b = -4, \quad c = 4
\][/tex]
Substituting these values into the discriminant formula, we get:
[tex]\[
\Delta = (-4)^2 - 4(\lambda)(4)
\][/tex]
Simplify the expression:
[tex]\[
\Delta = 16 - 16\lambda
\][/tex]
For the equation to have equal roots, the discriminant must be equal to zero:
[tex]\[
16 - 16\lambda = 0
\][/tex]
Solve for [tex]\(\lambda\)[/tex]:
[tex]\[
16 = 16\lambda
\][/tex]
[tex]\[
\lambda = 1
\][/tex]
Therefore, the value of [tex]\(\lambda\)[/tex] for which the equation [tex]\(\lambda x^2 - 4x + 4 = 0\)[/tex] has equal roots is [tex]\(\lambda = 1\)[/tex].
