Solve for x -1/2(x-3)^2=12

Solution:

Step 1

Apply the distributive property to eliminate the fraction by multiplying both sides by $-2$.

$$-2 \times \left(-\frac{1}{2}(x-3)^2\right) = -2 \times 12$$

Step 2

Perform simplification on both sides.

Step 2.1

Simplify the left-hand side.

Step 2.1.1

First, simplify the expression $-2 \times \left(-\frac{1}{2}(x-3)^2\right)$.

Step 2.1.1.1

Combine the squared term $(x-3)^2$ with $\frac{1}{2}$.

$$-2 \times \left(-\frac{(x-3)^2}{2}\right) = -2 \times 12$$

Step 2.1.1.2

Eliminate the common factor of $2$.

Step 2.1.1.2.1

Transfer the negative sign in $-\frac{(x-3)^2}{2}$ to the numerator.

$$-2 \times \frac{-(x-3)^2}{2} = -2 \times 12$$

Step 2.1.1.2.2

Extract $2$ from $-2$.

$$2 \times (-1) \times \frac{-(x-3)^2}{2} = -2 \times 12$$

Step 2.1.1.2.3

Cancel out the common factors.

$$\cancel{2} \times -1 \times \frac{-(x-3)^2}{\cancel{2}} = -2 \times 12$$

Step 2.1.1.2.4

Reformulate the expression.

$$(x-3)^2 = -2 \times 12$$

Step 2.1.1.3

Execute the multiplication.

$$(x-3)^2 = -24$$

Step 2.2

Simplify the right-hand side by multiplying $-2$ by $12$.

$$(x-3)^2 = -24$$

Step 3

Extract the square root on both sides to remove the exponent on the left side.

$$x - 3 = \pm \sqrt{-24}$$

Step 4

Simplify the square root of $-24$.

Step 4.1

Express $-24$ as $-1 \times 24$.

$$x - 3 = \pm \sqrt{-1 \times 24}$$

Step 4.2

Separate the square root of $-1$ and $24$.

$$x - 3 = \pm \sqrt{-1} \times \sqrt{24}$$

Step 4.3

Replace $\sqrt{-1}$ with the imaginary unit $i$.

$$x - 3 = \pm i \times \sqrt{24}$$

Step 4.4

Decompose $24$ into prime factors.

Step 4.4.1

Extract the square factor from $24$.

$$x - 3 = \pm i \times \sqrt{4 \times 6}$$

Step 4.4.2

Express $4$ as $2^2$.

$$x - 3 = \pm i \times \sqrt{2^2 \times 6}$$

Step 4.5

Extract square factors from under the radical.

$$x - 3 = \pm i \times (2 \sqrt{6})$$

Step 4.6

Reposition the $2$ in front of the imaginary unit $i$.

$$x - 3 = \pm 2i \sqrt{6}$$

Step 5

Determine the complete solution by considering both the positive and negative parts.

Step 5.1

Use the positive part of $\pm$ for the first solution.

$$x - 3 = 2i \sqrt{6}$$

Step 5.2

Add $3$ to both sides of the equation for the first solution.

$$x = 2i \sqrt{6} + 3$$

Step 5.3

Use the negative part of $\pm$ for the second solution.

$$x - 3 = -2i \sqrt{6}$$

Step 5.4

Add $3$ to both sides of the equation for the second solution.

$$x = -2i \sqrt{6} + 3$$

Step 5.5

Combine both solutions to express the complete solution.

$$x = 2i \sqrt{6} + 3, -2i \sqrt{6} + 3$$

Knowledge Notes:

The problem involves solving a quadratic equation with a complex solution. The key steps in solving such equations include:

1. Distributive Property: Used to eliminate fractions by multiplying both sides of the equation by a common denominator.

2. Simplification: Involves combining like terms and reducing expressions to their simplest form.

3. Square Roots: When dealing with squared terms, taking the square root of both sides is necessary to solve for the variable.

4. Complex Numbers: When the square root of a negative number is encountered, it is represented by the imaginary unit $i$, where $i^2 = -1$.

5. Radical Simplification: Breaking down a radical into its prime factors can sometimes simplify the expression, especially when square factors are present.

6. Solution of Quadratic Equations: Quadratic equations can have two solutions, which are found by considering both the positive and negative square roots.

In this problem, the quadratic equation had complex solutions because the term under the square root was negative, leading to the use of the imaginary unit $i$. The final solutions are expressed in terms of $i$ and real numbers.