Find all values of k for which the equation (k-2)x² - 2(k-1)x + k=0 has equal roots.
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Step-by-Step Solution
Step 1: Identify coefficients and condition for equal roots
For a quadratic equation in the standard form ax2+bx+c=0, the nature of its roots is determined by the discriminant, Δ=b2−4ac. If the roots are equal, the discriminant must be zero.
Step 2: Substitute coefficients into the discriminant formula
We identify the coefficients from the given equation: a=k−2, b=−2(k−1), and c=k. We substitute these values into the discriminant formula b2−4ac=0.
Step 3: Expand and simplify the equation
First, we expand the squared term and the product of the other two terms. Then, we distribute the 4 and combine like terms. This simplification leads to an unexpected result.
Step 4: Analyze the simplified equation
The equation simplifies to 4=0, which is a false statement. This means there are no values of k for which the discriminant is zero, given the initial assumption that the equation is quadratic.
Step 5: Consider the case where the equation is not quadratic
A quadratic equation requires the coefficient of x2 to be non-zero. If k−2=0, then k=2. In this case, the original equation becomes a linear equation, not a quadratic one.
Step 6: Check the linear case
If k=2, the equation becomes 0x2−2(1)x+2=0, which simplifies to −2x+2=0. Solving this linear equation gives x=1. A linear equation has exactly one root, which can be considered 'equal roots' in a degenerate sense.