Find All Complex Solutions (x-3)(x+2)=0

$(x - 3) (x + 2) = 0$

Step 1

Simplify the left side.

Simplify $(x - 3) (x + 2)$ .

Expand $(x - 3) (x + 2)$ using the FOIL Method.

Apply the distributive property.

$x (x + 2) - 3 (x + 2) = 0$

Apply the distributive property.

$x \cdot x + x \cdot 2 - 3 (x + 2) = 0$

Apply the distributive property.

$x \cdot x + x \cdot 2 - 3 x - 3 \cdot 2 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot 2 - 3 x - 3 \cdot 2 = 0$

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

$x^{2} + 2 \cdot x - 3 x - 3 \cdot 2 = 0$

Multiply $- 3$ by $2$ .

$x^{2} + 2 x - 3 x - 6 = 0$

Subtract $3 x$ from $2 x$ .

$x^{2} - x - 6 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = - 1$ , and $c = - 6$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot - 6)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 6}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 6$ .

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot - 6}}{2 \cdot 1}$

Multiply $- 4$ by $- 6$ .

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

Add $1$ and $24$ .

$x = \frac{1 \pm \sqrt{25}}{2 \cdot 1}$

Rewrite $25$ as $5^{2}$ .

$x = \frac{1 \pm \sqrt{5^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{1 \pm 5}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{1 \pm 5}{2}$

Step 5

The final answer is the combination of both solutions.

$x = 3, - 2$

Find All Complex Solutions (x-3)(x+2)=0

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