Understanding Equal Roots in Quadratic Equations

The question asks us to find the values of the parameter 'k' for the quadratic equation \(x^2 - (k-2)x + (k+1) = 0\) such that its roots are equal.

Quadratic Equation Basics

A standard quadratic equation is given in the form \(ax^2 + bx + c = 0\).

In our given equation, \(x^2 - (k-2)x + (k+1) = 0\), we can identify the coefficients:

• \(a = 1\)
• \(b = -(k-2)\)
• \(c = (k+1)\)

The Condition for Equal Roots

For a quadratic equation \(ax^2 + bx + c = 0\), the nature of its roots is determined by the discriminant, denoted by \(\Delta\) or \(D\). The formula for the discriminant is:

\( \Delta = b^2 - 4ac \)

The conditions based on the discriminant are:

• If \(\Delta > 0\), the equation has two distinct real roots.
• If \(\Delta = 0\), the equation has exactly one real root, or two equal real roots.
• If \(\Delta < 0\), the equation has two complex conjugate roots.

Since the question states that the roots are equal, we must have the discriminant equal to zero:

\( \Delta = 0 \)

Calculating the Discriminant

Now, let's substitute the coefficients \(a\), \(b\), and \(c\) from our specific equation into the discriminant formula:

\( \Delta = (-(k-2))^2 - 4(1)(k+1) \)

Solving for k

We set the discriminant to zero and solve for 'k':

\( (-(k-2))^2 - 4(1)(k+1) = 0 \)

Simplify the equation:

\( (k-2)^2 - 4(k+1) = 0 \)

Expand the terms:

\( (k^2 - 2(k)(2) + 2^2) - (4k + 4) = 0 \)

\( (k^2 - 4k + 4) - 4k - 4 = 0 \)

Combine like terms:

\( k^2 - 4k - 4k + 4 - 4 = 0 \)

\( k^2 - 8k = 0 \)

Factor out the common term 'k':

\( k(k - 8) = 0 \)

This equation gives two possible values for 'k':

• \(k = 0\)
• \(k - 8 = 0 \implies k = 8\)

Therefore, the values of 'k' for which the roots of the equation \(x^2 - (k-2)x + (k+1) = 0\) are equal are 0 and 8.