Find All Complex Solutions |x^2-2x|=24

$| x^{2} - 2 x | = 24$

Step 1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x^{2} - 2 x = \pm 24$

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x^{2} - 2 x = 24$

Subtract $24$ from both sides of the equation.

$x^{2} - 2 x - 24 = 0$

Factor $x^{2} - 2 x - 24$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Write the factored form using these integers.

$(x - 6) (x + 4) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 6 = 0$

$x + 4 = 0$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $(x - 6) (x + 4) = 0$ true.

$x = 6, - 4$

Next, use the negative value of the $\pm$ to find the second solution.

$x^{2} - 2 x = - 24$

Add $24$ to both sides of the equation.

$x^{2} - 2 x + 24 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 2$ , and $c = 24$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 24)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 24}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 24$ .

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 24}}{2 \cdot 1}$

Multiply $- 4$ by $24$ .

$x = \frac{2 \pm \sqrt{4 - 96}}{2 \cdot 1}$

Subtract $96$ from $4$ .

$x = \frac{2 \pm \sqrt{- 92}}{2 \cdot 1}$

Rewrite $- 92$ as $- 1 (92)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 92}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (92)}$ as $\sqrt{- 1} \cdot \sqrt{92}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{92}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{92}}{2 \cdot 1}$

Rewrite $92$ as $2^{2} \cdot 23$ .

Factor $4$ out of $92$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (23)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 23}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{23})}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{23}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i \sqrt{23}}{2}$

Simplify $\frac{2 \pm 2 i \sqrt{23}}{2}$ .

$x = 1 \pm i \sqrt{23}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 24}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 24$ .

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 24}}{2 \cdot 1}$

Multiply $- 4$ by $24$ .

$x = \frac{2 \pm \sqrt{4 - 96}}{2 \cdot 1}$

Subtract $96$ from $4$ .

$x = \frac{2 \pm \sqrt{- 92}}{2 \cdot 1}$

Rewrite $- 92$ as $- 1 (92)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 92}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (92)}$ as $\sqrt{- 1} \cdot \sqrt{92}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{92}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{92}}{2 \cdot 1}$

Rewrite $92$ as $2^{2} \cdot 23$ .

Factor $4$ out of $92$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (23)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 23}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{23})}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{23}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i \sqrt{23}}{2}$

Simplify $\frac{2 \pm 2 i \sqrt{23}}{2}$ .

$x = 1 \pm i \sqrt{23}$

Change the $\pm$ to $+$ .

$x = 1 + i \sqrt{23}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 24}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 24$ .

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 24}}{2 \cdot 1}$

Multiply $- 4$ by $24$ .

$x = \frac{2 \pm \sqrt{4 - 96}}{2 \cdot 1}$

Subtract $96$ from $4$ .

$x = \frac{2 \pm \sqrt{- 92}}{2 \cdot 1}$

Rewrite $- 92$ as $- 1 (92)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 92}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (92)}$ as $\sqrt{- 1} \cdot \sqrt{92}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{92}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{92}}{2 \cdot 1}$

Rewrite $92$ as $2^{2} \cdot 23$ .

Factor $4$ out of $92$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (23)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 23}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{23})}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{23}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i \sqrt{23}}{2}$

Simplify $\frac{2 \pm 2 i \sqrt{23}}{2}$ .

$x = 1 \pm i \sqrt{23}$

Change the $\pm$ to $-$ .

$x = 1 - i \sqrt{23}$

The final answer is the combination of both solutions.

$x = 1 + i \sqrt{23}, 1 - i \sqrt{23}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 6, - 4, 1 + i \sqrt{23}, 1 - i \sqrt{23}$

Find All Complex Solutions |x^2-2x|=24

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