The solution to 3×2 – 12x 24 = 0 is
The Correct Answer and Explanation is :
The equation (3x^2 – 12x + 24 = 0) is a quadratic equation. To solve it, we’ll follow these steps:
1. Simplify the Equation
Divide the entire equation by 3 to simplify:
[
x^2 – 4x + 8 = 0
]
2. Calculate the Discriminant ((\Delta))
The discriminant of a quadratic equation (ax^2 + bx + c = 0) is given by:
[
\Delta = b^2 – 4ac
]
Here, (a = 1), (b = -4), and (c = 8). Substituting:
[
\Delta = (-4)^2 – 4(1)(8) = 16 – 32 = -16
]
Since the discriminant is negative ((\Delta < 0)), the equation has no real solutions, only complex solutions.
3. Solve Using the Quadratic Formula
The quadratic formula is:
[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
]
Substitute the values of (a), (b), and (\Delta):
[
x = \frac{-(-4) \pm \sqrt{-16}}{2(1)} = \frac{4 \pm \sqrt{-16}}{2} = \frac{4 \pm 4i}{2}
]
Simplify:
[
x = 2 \pm 2i
]
Final Answer:
The solutions are:
[
x = 2 + 2i \quad \text{and} \quad x = 2 – 2i
]
Explanation:
- The quadratic equation represents a parabola.
- A negative discriminant means the parabola does not intersect the (x)-axis, resulting in complex solutions.
- Complex solutions include both a real part (2) and an imaginary part ((\pm 2i)).
- Solving with the quadratic formula ensures we account for all possible roots, including complex numbers. These roots represent solutions in the complex plane rather than on the real number line.