Evaluate f(x)=2 square root of -9

Solution:

Step 1: Simplify the expression $2\sqrt{-9}$.

Step 1.1: Express the function as $f(x)=0+0+2\sqrt{-9}$.

Step 1.2: Combine like terms to simplify $f(x)=2\sqrt{-9}$.

Step 1.3: Factor -9 as $-1(9)$ to get $f(x)=2\sqrt{-1(9)}$.

Step 1.4: Separate the square root of the product into the product of square roots, yielding $f(x)=2(\sqrt{-1}\cdot \sqrt{9})$.

Step 1.5: Replace $\sqrt{-1}$ with the imaginary unit $i$, so $f(x)=2(i\sqrt{9})$.

Step 1.6: Express 9 as $3^2$ to get $f(x)=2(i\sqrt{3^2})$.

Step 1.7: Extract the square root of the perfect square to obtain $f(x)=2(i\cdot 3)$.

Step 1.8: Rearrange the terms to place the constant before the imaginary unit, resulting in $f(x)=2(3i)$.

Step 1.9: Multiply 3 by 2 to finalize the simplification as $f(x)=6i$.

Step 2: Isolate the function $f(x)$ on one side of the equation.

Step 2.1: Subtract $f(x)$ from both sides to get $0=6i-f(x)$.

Step 2.2: Rearrange the terms to form $0=-f(x)+6i$.

Step 3: Rewrite the equation in standard form.

Step 3.1: Present the equation as $-f(x)+6i=0$.

Knowledge Notes:

1. Imaginary Numbers: Imaginary numbers are an extension of the real number system where the square root of a negative number is defined. The imaginary unit is denoted as $i$, where $i^2=-1$.

2. Square Roots: The square root of a number $x$ is a value that, when multiplied by itself, gives $x$. For positive real numbers, the square root is also real. However, for negative numbers, the square root is imaginary.

3. Simplification: Simplifying an expression involves combining like terms and reducing the expression to its simplest form. This can include factoring, expanding, and canceling terms.

4. Complex Numbers: Complex numbers consist of a real part and an imaginary part, expressed as $a+bi$ where $a$ and $b$ are real numbers. In this problem, the result is purely imaginary since the real part is zero.

5. Radical Properties: The square root of a product can be expressed as the product of the square roots of the factors. Additionally, the square root of a perfect square, such as $9$, is a rational number ($\sqrt{9}=3$).

6. Equation Rearrangement: Rearranging an equation involves moving terms from one side of the equation to the other, often to isolate a variable or to express the equation in a standard form.

7. Multiplication of Imaginary Numbers: When multiplying an imaginary number by a real number, the result is an imaginary number. The multiplication follows the commutative property, allowing the rearrangement of terms.