How can I solve the equation x^2 + 4x + 12 = 0, and how can I simplify the expression x^2 × 6 × 2 × 6 × 3 × 4 ?

Solving the Equation x² + 4x + 12 = 0

The equation you provided is a quadratic equation in the standard form Ax² + Bx + C = 0, where A = 1, B = 4, and C = 12.

To solve this equation, we can use the quadratic formula:

x = (-B ± √(B² – 4AC)) / (2A)

Substituting the values of A, B, and C, we get:

B² – 4AC = 4² – 4 × 1 × 12 = 16 – 48 = -32

Since the discriminant (B² – 4AC) is negative, this quadratic equation has no real solutions; instead, it has two complex solutions.

We can express these solutions as follows:

x = (-4 ± √(-32)) / 2

We can further simplify it:

√(-32) = √(32) × √(-1) = 4√(2)i

Thus, the solutions are:

x = (-4 + 4√(2)i) / 2 = -2 + 2√(2)i

and

x = (-4 – 4√(2)i) / 2 = -2 – 2√(2)i

Now, let’s simplify the expression:

x² × 6 × 2 × 6 × 3 × 4

First, we can multiply the constants:

6 × 2 = 12

12 × 6 = 72

72 × 3 = 216

216 × 4 = 864

Now we can write the expression as:

x² × 864

So, the simplified expression is:

864x²