Let's solve the quadratic equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex] step by step.
1. Rewrite the equation:
The equation we need to solve is:
[tex]\[ x^2 - 2x - 24 = 0 \][/tex]
2. Identify the coefficients:
For a quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex], we identify [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -24 \)[/tex].
3. Use the quadratic formula:
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
4. Plug in the coefficients:
[tex]\[ a = 1, \, b = -2, \, c = -24 \][/tex]
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1} \][/tex]
5. Simplify inside the square root:
[tex]\[ x = \frac{2 \pm \sqrt{4 + 96}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{100}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm 10}{2} \][/tex]
6. Find the two potential solutions by splitting the expression:
[tex]\[
\text{Solution 1: } x = \frac{2 + 10}{2} = \frac{12}{2} = 6
\][/tex]
[tex]\[
\text{Solution 2: } x = \frac{2 - 10}{2} = \frac{-8}{2} = -4
\][/tex]
7. List the solutions:
The solutions to the equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex] are [tex]\( x = 6 \)[/tex] and [tex]\( x = -4 \)[/tex].
8. Match the solutions with the given choices:
- A. -6
- B. -24
- C. -4 (Correct)
- D. 4
- E. 6 (Correct)
Therefore, the correct solutions to the equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex] are:
- C. -4
- E. 6
