Trigonometry Examples

${(x - y)}^{2} + 2 x y - 3 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = 0$ , and $c = y^{2} - 3$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (1 \cdot (y^{2} - 3))}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raising $0$ to any positive power yields $0$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 1 \cdot (y^{2} - 3)}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4$ by $1$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot (y^{2} - 3)}}{2 \cdot 1}$

Step 3.1.3

Apply the distributive property.

$x = \frac{0 \pm \sqrt{0 - 4 y^{2} - 4 \cdot - 3}}{2 \cdot 1}$

Step 3.1.4

Multiply $- 4$ by $- 3$ .

$x = \frac{0 \pm \sqrt{0 - 4 y^{2} + 12}}{2 \cdot 1}$

Step 3.1.5

Subtract $- (- 4 y^{2} + 12)$ from $0$ .

$x = \frac{0 \pm \sqrt{- 4 y^{2} + 12}}{2 \cdot 1}$

Step 3.1.6

Factor $4$ out of $- 4 y^{2} + 12$ .

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Step 3.1.6.1

Factor $4$ out of $- 4 y^{2}$ .

$x = \frac{0 \pm \sqrt{4 (- y^{2}) + 12}}{2 \cdot 1}$

Step 3.1.6.2

Factor $4$ out of $12$ .

$x = \frac{0 \pm \sqrt{4 (- y^{2}) + 4 (3)}}{2 \cdot 1}$

Step 3.1.6.3

Factor $4$ out of $4 (- y^{2}) + 4 (3)$ .

$x = \frac{0 \pm \sqrt{4 (- y^{2} + 3)}}{2 \cdot 1}$

Step 3.1.7

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

$x = \frac{0 \pm \sqrt{2^{2} (- y^{2} + 3)}}{2 \cdot 1}$

Step 3.1.8

Pull terms out from under the radical.

$x = \frac{0 \pm 2 \sqrt{- y^{2} + 3}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$x = \frac{0 \pm 2 \sqrt{- y^{2} + 3}}{2}$

Step 3.3

Simplify $\frac{0 \pm 2 \sqrt{- y^{2} + 3}}{2}$ .

$x = \pm \sqrt{- y^{2} + 3}$

Step 4

The final answer is the combination of both solutions.

$x = \sqrt{- y^{2} + 3}$

$x = - \sqrt{- y^{2} + 3}$