Finding the value of m for two real and equal roots
To find the value of m for which the quadratic equation has two real and equal roots, we need to use the discriminant of the quadratic formula.

Given Quadratic Equation:
(m − 1)x^2 + 2(m − 1)x + 1 = 0

Discriminant Formula:
The discriminant (Δ) of a quadratic equation ax^2 + bx + c = 0 is given by Δ = b^2 - 4ac. If Δ = 0, the equation has two real and equal roots.

Substitute the Coefficients:
In this case, a = m - 1, b = 2(m - 1), and c = 1.
Substitute these values into the discriminant formula: Δ = (2(m - 1))^2 - 4(m - 1)(1).

Calculate the Discriminant:
Expand the equation: Δ = 4(m^2 - 2m + 1) - 4m + 4
Simplify the expression: Δ = 4m^2 - 8m + 4 - 4m + 4
Combine like terms: Δ = 4m^2 - 12m + 8

Set Discriminant to Zero:
For the equation to have two real and equal roots, the discriminant should be zero.
So, set Δ = 0 and solve the equation: 4m^2 - 12m + 8 = 0

Find the Value of m:
Now, solve the quadratic equation 4m^2 - 12m + 8 = 0 to find the value of m.
You can use the quadratic formula or factorization to solve for m.

Conclusion:
By setting the discriminant to zero and solving the resulting quadratic equation, you can find the value of m for which the given quadratic equation has two real and equal roots.