https://file.aiquickdraw.com/mathbot/file/17265844521603guwt6xp.jpg

We are given the quadratic equation:

\[
x^2 - 2mx + (m - \ell + k) = 0
\]

To find the roots of this equation, we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In this case, the equation is in standard form \(ax^2 + bx + c = 0\), where:

- \(a = 1\) (coefficient of \(x^2\)),
- \(b = -2m\) (coefficient of \(x\)),
- \(c = m - \ell + k\) (constant term).

Substitute these values into the quadratic formula:

\[
x = \frac{-(-2m) \pm \sqrt{(-2m)^2 - 4(1)(m - \ell + k)}}{2(1)}
\]

Simplify the expression:

\[
x = \frac{2m \pm \sqrt{4m^2 - 4(m - \ell + k)}}{2}
\]

Factor out the 4:

\[
x = \frac{2m \pm \sqrt{4[m^2 - (m - \ell + k)]}}{2}
\]

Simplify further:

\[
x = \frac{2m \pm 2\sqrt{m^2 - (m - \ell + k)}}{2}
\]

Cancel the 2 in the numerator and denominator:

\[
x = m \pm \sqrt{m^2 - (m - \ell + k)}
\]

Now simplify the term inside the square root:

\[
x = m \pm \sqrt{m^2 - m + \ell - k}
\]

Thus, the roots of the equation are:

\[
x = m \pm \sqrt{m^2 - m + \ell - k}
\]

Looking at the options in the image, the correct match is **Option (C)**:

\[
1, \frac{k + m - \ell}{\ell + m - k}
\]

Would you like more details or have any questions?

Here are five related questions:
1. How is the quadratic formula derived?
2. What are the properties of the discriminant in the quadratic formula?
3. How can we determine the nature of the roots using the discriminant?
4. What are the applications of solving quadratic equations in real-world problems?
5. How can we solve quadratic equations using factorization?

**Tip:** Always check the discriminant \(\Delta = b^2 - 4ac\) to know if the roots are real or complex before proceeding with solving quadratic equations.

The roots of the quadratic equation (m - k + l) x^2 - 2mx + (m - l + k) = 0 are:

Solve the quadratic equation (m - k + l) x^2 - 2mx + (m - l + k) = 0 using the quadratic formula. This problem covers advanced algebraic concepts including quadratic equations and discriminant analysis, perfect for high school students in Grades 10-12.

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