To solve the quadratic equation \( x^2 - 24x + 140 = 0 \), we can use the quadratic formula:

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

For the given equation, the coefficients are:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -24 \) (coefficient of \( x \))
- \( c = 140 \) (constant term)

Now, let's plug in these values into the quadratic formula:

\[
x = \frac{-(-24) \pm \sqrt{(-24)^2 - 4(1)(140)}}{2(1)}
\]

Simplifying the expression:

\[
x = \frac{24 \pm \sqrt{576 - 560}}{2}
\]

\[
x = \frac{24 \pm \sqrt{16}}{2}
\]

\[
x = \frac{24 \pm 4}{2}
\]

This gives us two possible solutions:

\[
x_1 = \frac{24 + 4}{2} = \frac{28}{2} = 14
\]

\[
x_2 = \frac{24 - 4}{2} = \frac{20}{2} = 10
\]

So, the solutions are \( x = 14 \) and \( x = 10 \).

Would you like more details or have any questions?

Here are some related questions:
1. What are the steps to derive the quadratic formula?
2. How do you determine the nature of the roots using the discriminant?
3. What is the vertex form of a quadratic equation, and how can you convert it?
4. How do you solve a quadratic equation by completing the square?
5. What are some applications of quadratic equations in real life?

**Tip:** Always check your solutions by substituting them back into the original equation to verify they satisfy it.

Learn how to solve the quadratic equation x^2 - 24x + 140 = 0 with a detailed step-by-step solution. This problem involves finding roots using the quadratic formula and applies algebraic concepts such as coefficients and constants. Suitable for high school students (Grades 9-12).

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