convert from standard form to factored form and highlight your final factored form. then determine the solutions to the equation: x^2 - 4x + 24

To convert \( x^2 - 4x + 24 \) from standard form to factored form, we need to find factors of the quadratic expression. 

### Step 1: Identify the quadratic equation
The quadratic equation given is:
\[
x^2 - 4x + 24 = 0
\]

### Step 2: Calculate the discriminant
To determine whether the quadratic equation can be factored over real numbers, we calculate the discriminant, \( \Delta \), using the formula:
\[
\Delta = b^2 - 4ac
\]

For \( x^2 - 4x + 24 \), \( a = 1 \), \( b = -4 \), and \( c = 24 \). Plug these values in:
\[
\Delta = (-4)^2 - 4(1)(24) = 16 - 96 = -80
\]

Since the discriminant is negative, it indicates that the equation has complex roots and therefore cannot be factored over the real numbers.

### Step 3: Determine the roots using the quadratic formula
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Plugging in the values:
\[
x = \frac{-(-4) \pm \sqrt{-80}}{2(1)} = \frac{4 \pm \sqrt{-80}}{2}
\]

Express \( \sqrt{-80} \) as \( \sqrt{-1 \cdot 16 \cdot 5} = 4i\sqrt{5} \). Now substitute:
\[
x = \frac{4 \pm 4i\sqrt{5}}{2}
\]

This simplifies to:
\[
x = 2 \pm 2i\sqrt{5}
\]

### Step 4: Express in factored form
Since the solutions are \( x = 2 + 2i\sqrt{5} \) and \( x = 2 - 2i\sqrt{5} \), we can write the quadratic in factored form as:
\[
(x - (2 + 2i\sqrt{5}))(x - (2 - 2i\sqrt{5}))
\]

### Final Answer (highlighted):
The **factored form** of the equation is:
\[
(x - (2 + 2i\sqrt{5}))(x - (2 - 2i\sqrt{5}))
\]

The **solutions** to the equation are:
\[
x = 2 + 2i\sqrt{5} \quad \text{and} \quad x = 2 - 2i\sqrt{5}
\]

Would you like more details or have any questions?

### Related Questions:
1. How can we determine the number of real solutions for a quadratic equation?
2. What is the significance of a negative discriminant in quadratic equations?
3. How do you convert factored form back to standard form?
4. Can quadratic equations always be factored over the real numbers?
5. What are some applications of quadratic equations with complex solutions?

**Tip:** When the discriminant of a quadratic equation is negative, it indicates the presence of complex roots.

Learn how to convert the quadratic equation x^2 - 4x + 24 from standard form to factored form and find its solutions. This solution uses key algebraic techniques, including the discriminant theorem and the quadratic formula, to identify complex roots. Suitable for students in Grades 10-12.

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