Linear Algebra Examples

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

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 $a = 1$ , $b = 4$ , and $c = - 4 x - 4$ into the quadratic formula and solve for $y$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $4$ to the power of $2$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (- 4 x - 4)}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot (- 4 x - 4)}}{2 \cdot 1}$

Step 3.1.3

Apply the distributive property.

$y = \frac{- 4 \pm \sqrt{16 - 4 (- 4 x) - 4 \cdot - 4}}{2 \cdot 1}$

Step 3.1.4

Multiply $- 4$ by $- 4$ .

$y = \frac{- 4 \pm \sqrt{16 + 16 x - 4 \cdot - 4}}{2 \cdot 1}$

Step 3.1.5

Multiply $- 4$ by $- 4$ .

$y = \frac{- 4 \pm \sqrt{16 + 16 x + 16}}{2 \cdot 1}$

Step 3.1.6

Add $16$ and $16$ .

$y = \frac{- 4 \pm \sqrt{16 x + 32}}{2 \cdot 1}$

Step 3.1.7

Factor $16$ out of $16 x + 32$ .

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Step 3.1.7.1

Factor $16$ out of $16 x$ .

$y = \frac{- 4 \pm \sqrt{16 (x) + 32}}{2 \cdot 1}$

Step 3.1.7.2

Factor $16$ out of $32$ .

$y = \frac{- 4 \pm \sqrt{16 x + 16 \cdot 2}}{2 \cdot 1}$

Step 3.1.7.3

Factor $16$ out of $16 x + 16 \cdot 2$ .

$y = \frac{- 4 \pm \sqrt{16 (x + 2)}}{2 \cdot 1}$

Step 3.1.8

Rewrite $16 (x + 2)$ as ${(2^{2})}^{2} (x + 2)$ .

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Step 3.1.8.1

Rewrite $16$ as $4^{2}$ .

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

Step 3.1.8.2

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

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

Step 3.1.9

Pull terms out from under the radical.

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

Step 3.1.10

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

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

Step 3.2

Multiply $2$ by $1$ .

$y = \frac{- 4 \pm 4 \sqrt{x + 2}}{2}$

Step 3.3

Simplify $\frac{- 4 \pm 4 \sqrt{x + 2}}{2}$ .

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

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $4$ to the power of $2$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (- 4 x - 4)}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot (- 4 x - 4)}}{2 \cdot 1}$

Step 4.1.3

Apply the distributive property.

$y = \frac{- 4 \pm \sqrt{16 - 4 (- 4 x) - 4 \cdot - 4}}{2 \cdot 1}$

Step 4.1.4

Multiply $- 4$ by $- 4$ .

$y = \frac{- 4 \pm \sqrt{16 + 16 x - 4 \cdot - 4}}{2 \cdot 1}$

Step 4.1.5

Multiply $- 4$ by $- 4$ .

$y = \frac{- 4 \pm \sqrt{16 + 16 x + 16}}{2 \cdot 1}$

Step 4.1.6

Add $16$ and $16$ .

$y = \frac{- 4 \pm \sqrt{16 x + 32}}{2 \cdot 1}$

Step 4.1.7

Factor $16$ out of $16 x + 32$ .

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Step 4.1.7.1

Factor $16$ out of $16 x$ .

$y = \frac{- 4 \pm \sqrt{16 (x) + 32}}{2 \cdot 1}$

Step 4.1.7.2

Factor $16$ out of $32$ .

$y = \frac{- 4 \pm \sqrt{16 x + 16 \cdot 2}}{2 \cdot 1}$

Step 4.1.7.3

Factor $16$ out of $16 x + 16 \cdot 2$ .

$y = \frac{- 4 \pm \sqrt{16 (x + 2)}}{2 \cdot 1}$

Step 4.1.8

Rewrite $16 (x + 2)$ as ${(2^{2})}^{2} (x + 2)$ .

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Step 4.1.8.1

Rewrite $16$ as $4^{2}$ .

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

Step 4.1.8.2

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

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

Step 4.1.9

Pull terms out from under the radical.

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

Step 4.1.10

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

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

Step 4.2

Multiply $2$ by $1$ .

$y = \frac{- 4 \pm 4 \sqrt{x + 2}}{2}$

Step 4.3

Simplify $\frac{- 4 \pm 4 \sqrt{x + 2}}{2}$ .

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

Step 4.4

Change the $\pm$ to $+$ .

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

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $4$ to the power of $2$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (- 4 x - 4)}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot (- 4 x - 4)}}{2 \cdot 1}$

Step 5.1.3

Apply the distributive property.

$y = \frac{- 4 \pm \sqrt{16 - 4 (- 4 x) - 4 \cdot - 4}}{2 \cdot 1}$

Step 5.1.4

Multiply $- 4$ by $- 4$ .

$y = \frac{- 4 \pm \sqrt{16 + 16 x - 4 \cdot - 4}}{2 \cdot 1}$

Step 5.1.5

Multiply $- 4$ by $- 4$ .

$y = \frac{- 4 \pm \sqrt{16 + 16 x + 16}}{2 \cdot 1}$

Step 5.1.6

Add $16$ and $16$ .

$y = \frac{- 4 \pm \sqrt{16 x + 32}}{2 \cdot 1}$

Step 5.1.7

Factor $16$ out of $16 x + 32$ .

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Step 5.1.7.1

Factor $16$ out of $16 x$ .

$y = \frac{- 4 \pm \sqrt{16 (x) + 32}}{2 \cdot 1}$

Step 5.1.7.2

Factor $16$ out of $32$ .

$y = \frac{- 4 \pm \sqrt{16 x + 16 \cdot 2}}{2 \cdot 1}$

Step 5.1.7.3

Factor $16$ out of $16 x + 16 \cdot 2$ .

$y = \frac{- 4 \pm \sqrt{16 (x + 2)}}{2 \cdot 1}$

Step 5.1.8

Rewrite $16 (x + 2)$ as ${(2^{2})}^{2} (x + 2)$ .

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Step 5.1.8.1

Rewrite $16$ as $4^{2}$ .

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

Step 5.1.8.2

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

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

Step 5.1.9

Pull terms out from under the radical.

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

Step 5.1.10

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

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

Step 5.2

Multiply $2$ by $1$ .

$y = \frac{- 4 \pm 4 \sqrt{x + 2}}{2}$

Step 5.3

Simplify $\frac{- 4 \pm 4 \sqrt{x + 2}}{2}$ .

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

Step 5.4

Change the $\pm$ to $-$ .

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

Step 6

The final answer is the combination of both solutions.

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

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

Step 7

Set the radicand in $\sqrt{x + 2}$ greater than or equal to $0$ to find where the expression is defined.

$x + 2 \geq 0$

Step 8

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

Step 9

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- 2, \infty)$

Set-Builder Notation:

${x | x \geq - 2}$

Step 10