Algebra Examples

$y^{2} + 6 y + 9 = x + 8$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Move all the expressions to the left side of the equation.

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Step 1.1.1

Subtract $x$ from both sides of the equation.

$y^{2} + 6 y + 9 - x = 8$

Step 1.1.2

Subtract $8$ from both sides of the equation.

$y^{2} + 6 y + 9 - x - 8 = 0$

Step 1.2

Subtract $8$ from $9$ .

$y^{2} + 6 y - x + 1 = 0$

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $6$ to the power of $2$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot (- x + 1)}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot (- x + 1)}}{2 \cdot 1}$

Step 4.1.3

Apply the distributive property.

$y = \frac{- 6 \pm \sqrt{36 - 4 (- x) - 4 \cdot 1}}{2 \cdot 1}$

Step 4.1.4

Multiply $- 1$ by $- 4$ .

$y = \frac{- 6 \pm \sqrt{36 + 4 x - 4 \cdot 1}}{2 \cdot 1}$

Step 4.1.5

Multiply $- 4$ by $1$ .

$y = \frac{- 6 \pm \sqrt{36 + 4 x - 4}}{2 \cdot 1}$

Step 4.1.6

Subtract $4$ from $36$ .

$y = \frac{- 6 \pm \sqrt{4 x + 32}}{2 \cdot 1}$

Step 4.1.7

Factor $4$ out of $4 x + 32$ .

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Step 4.1.7.1

Factor $4$ out of $4 x$ .

$y = \frac{- 6 \pm \sqrt{4 (x) + 32}}{2 \cdot 1}$

Step 4.1.7.2

Factor $4$ out of $32$ .

$y = \frac{- 6 \pm \sqrt{4 x + 4 \cdot 8}}{2 \cdot 1}$

Step 4.1.7.3

Factor $4$ out of $4 x + 4 \cdot 8$ .

$y = \frac{- 6 \pm \sqrt{4 (x + 8)}}{2 \cdot 1}$

Step 4.1.8

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

$y = \frac{- 6 \pm \sqrt{2^{2} (x + 8)}}{2 \cdot 1}$

Step 4.1.9

Pull terms out from under the radical.

$y = \frac{- 6 \pm 2 \sqrt{x + 8}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$y = \frac{- 6 \pm 2 \sqrt{x + 8}}{2}$

Step 4.3

Simplify $\frac{- 6 \pm 2 \sqrt{x + 8}}{2}$ .

$y = - 3 \pm \sqrt{x + 8}$

Step 5

The final answer is the combination of both solutions.

$y = - 3 + \sqrt{x + 8}$

$y = - 3 - \sqrt{x + 8}$