Use all the methods to solve the quadratic equations 

Sure, I'll demonstrate solving quadratic equations using three methods: factoring, completing the square, and the quadratic formula.

### Quadratic Equation: \( x^2 - 10x - 24 = 0 \)

#### 1. Factoring

First, let's solve by factoring:

\[
x^2 - 10x - 24 = 0
\]

We need to find two numbers that multiply to \(-24\) and add to \(-10\). These numbers are \(-12\) and \(2\).

So, we factorize:

\[
(x - 12)(x + 2) = 0
\]

Setting each factor to zero gives:

\[
x - 12 = 0 \quad \text{or} \quad x + 2 = 0
\]

\[
x = 12 \quad \text{or} \quad x = -2
\]

#### 2. Completing the Square

Next, let's solve by completing the square:

\[
x^2 - 10x - 24 = 0
\]

Move the constant term to the other side:

\[
x^2 - 10x = 24
\]

To complete the square, add \(\left(\frac{-10}{2}\right)^2 = 25\) to both sides:

\[
x^2 - 10x + 25 = 24 + 25
\]

\[
(x - 5)^2 = 49
\]

Take the square root of both sides:

\[
x - 5 = \pm 7
\]

So, we have two solutions:

\[
x = 5 + 7 = 12 \quad \text{or} \quad x = 5 - 7 = -2
\]

#### 3. Quadratic Formula

Finally, let's use the quadratic formula:

\[
ax^2 + bx + c = 0 \quad \text{with} \quad a = 1, \quad b = -10, \quad c = -24
\]

The quadratic formula is:

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

Plugging in the values:

\[
x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(-24)}}{2(1)}
\]

\[
x = \frac{10 \pm \sqrt{100 + 96}}{2}
\]

\[
x = \frac{10 \pm \sqrt{196}}{2}
\]

\[
x = \frac{10 \pm 14}{2}
\]

So, we have two solutions:

\[
x = \frac{10 + 14}{2} = 12 \quad \text{or} \quad x = \frac{10 - 14}{2} = -2
\]

### Summary

We used three different methods to solve the quadratic equation \(x^2 - 10x - 24 = 0\), and in each case, the solutions are:

\[
x = 12 \quad \text{or} \quad x = -2
\]

Would you like to see another example solved using all three methods? Here are some related questions to consider:

1. How do you determine which method to use for solving a quadratic equation?
2. What are the advantages and disadvantages of each method?
3. How do you recognize a perfect square trinomial?
4. Can completing the square be used for all quadratic equations?
5. How can the quadratic formula be derived?
6. What do the discriminant and its value tell you about the nature of the roots?
7. How do you factorize a quadratic equation when the leading coefficient is not 1?
8. What is the significance of the vertex form of a quadratic equation?

**Tip:** Practice solving quadratic equations using all three methods to become proficient and recognize which method might be the most efficient for different types of equations.

Explore how to solve quadratic equations using three methods: factoring, completing the square, and the quadratic formula. Step-by-step solutions are provided for the quadratic equation x^2 - 10x - 24 = 0, covering key algebraic concepts and formulas.

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