Trigonometry Examples

$3 x^{2} - x - 3 = x^{2} - 5 x - 7$

Step 1

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 1.1

Subtract $x^{2}$ from both sides of the equation.

$3 x^{2} - x - 3 - x^{2} = - 5 x - 7$

Step 1.2

Add $5 x$ to both sides of the equation.

$3 x^{2} - x - 3 - x^{2} + 5 x = - 7$

Step 1.3

Subtract $x^{2}$ from $3 x^{2}$ .

$2 x^{2} - x - 3 + 5 x = - 7$

Step 1.4

Add $- x$ and $5 x$ .

$2 x^{2} + 4 x - 3 = - 7$

Step 2

Add $7$ to both sides of the equation.

$2 x^{2} + 4 x - 3 + 7 = 0$

Step 3

Add $- 3$ and $7$ .

$2 x^{2} + 4 x + 4 = 0$

Step 4

Factor $2$ out of $2 x^{2} + 4 x + 4$ .

Tap for more steps...

Step 4.1

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) + 4 x + 4 = 0$

Step 4.2

Factor $2$ out of $4 x$ .

$2 (x^{2}) + 2 (2 x) + 4 = 0$

Step 4.3

Factor $2$ out of $4$ .

$2 x^{2} + 2 (2 x) + 2 \cdot 2 = 0$

Step 4.4

Factor $2$ out of $2 x^{2} + 2 (2 x)$ .

$2 (x^{2} + 2 x) + 2 \cdot 2 = 0$

Step 4.5

Factor $2$ out of $2 (x^{2} + 2 x) + 2 \cdot 2$ .

$2 (x^{2} + 2 x + 2) = 0$

Step 5

Divide each term in $2 (x^{2} + 2 x + 2) = 0$ by $2$ and simplify.

Tap for more steps...

Step 5.1

Divide each term in $2 (x^{2} + 2 x + 2) = 0$ by $2$ .

$\frac{2 (x^{2} + 2 x + 2)}{2} = \frac{0}{2}$

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} + 2 x + 2)}{2} = \frac{0}{2}$

Step 5.2.1.2

Divide $x^{2} + 2 x + 2$ by $1$ .

$x^{2} + 2 x + 2 = \frac{0}{2}$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Divide $0$ by $2$ .

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

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

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

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

Step 8

Simplify.

Tap for more steps...

Step 8.1

Simplify the numerator.

Tap for more steps...

Step 8.1.1

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

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

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 8.1.2.1

Multiply $- 4$ by $1$ .

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

Step 8.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 8.1.3

Subtract $8$ from $4$ .

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

Step 8.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 8.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 8.1.6

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

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

Step 8.1.7

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

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

Step 8.1.8

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

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

Step 8.1.9

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

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

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 i}{2}$

Step 8.3

Simplify $\frac{- 2 \pm 2 i}{2}$ .

$x = - 1 \pm i$

Step 9

The final answer is the combination of both solutions.

$x = - 1 + i, - 1 - i$