Algebra Examples

$8 x + y^{2} - 2 y = 64 - y^{2}$

Step 1

Solve for $y$ .

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Step 1.1

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

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Step 1.1.1

Add $y^{2}$ to both sides of the equation.

$8 x + y^{2} - 2 y + y^{2} = 64$

Step 1.1.2

Add $y^{2}$ and $y^{2}$ .

$8 x + 2 y^{2} - 2 y = 64$

Step 1.2

Subtract $64$ from both sides of the equation.

$8 x + 2 y^{2} - 2 y - 64 = 0$

Step 1.3

Use the quadratic formula to find the solutions.

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

Step 1.4

Substitute the values $a = 2$ , $b = - 2$ , and $c = 8 x - 64$ into the quadratic formula and solve for $y$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (2 \cdot (8 x - 64))}}{2 \cdot 2}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot (8 x - 64)}}{2 \cdot 2}$

Step 1.5.1.2

Multiply $- 4$ by $2$ .

$y = \frac{2 \pm \sqrt{4 - 8 \cdot (8 x - 64)}}{2 \cdot 2}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{2 \pm \sqrt{4 - 8 (8 x) - 8 \cdot - 64}}{2 \cdot 2}$

Step 1.5.1.4

Multiply $8$ by $- 8$ .

$y = \frac{2 \pm \sqrt{4 - 64 x - 8 \cdot - 64}}{2 \cdot 2}$

Step 1.5.1.5

Multiply $- 8$ by $- 64$ .

$y = \frac{2 \pm \sqrt{4 - 64 x + 512}}{2 \cdot 2}$

Step 1.5.1.6

Add $4$ and $512$ .

$y = \frac{2 \pm \sqrt{- 64 x + 516}}{2 \cdot 2}$

Step 1.5.1.7

Factor $4$ out of $- 64 x + 516$ .

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Step 1.5.1.7.1

Factor $4$ out of $- 64 x$ .

$y = \frac{2 \pm \sqrt{4 (- 16 x) + 516}}{2 \cdot 2}$

Step 1.5.1.7.2

Factor $4$ out of $516$ .

$y = \frac{2 \pm \sqrt{4 (- 16 x) + 4 (129)}}{2 \cdot 2}$

Step 1.5.1.7.3

Factor $4$ out of $4 (- 16 x) + 4 (129)$ .

$y = \frac{2 \pm \sqrt{4 (- 16 x + 129)}}{2 \cdot 2}$

Step 1.5.1.8

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

$y = \frac{2 \pm \sqrt{2^{2} (- 16 x + 129)}}{2 \cdot 2}$

Step 1.5.1.9

Pull terms out from under the radical.

$y = \frac{2 \pm 2 \sqrt{- 16 x + 129}}{2 \cdot 2}$

Step 1.5.2

Multiply $2$ by $2$ .

$y = \frac{2 \pm 2 \sqrt{- 16 x + 129}}{4}$

Step 1.5.3

Simplify $\frac{2 \pm 2 \sqrt{- 16 x + 129}}{4}$ .

$y = \frac{1 \pm \sqrt{- 16 x + 129}}{2}$

Step 1.6

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

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot (8 x - 64)}}{2 \cdot 2}$

Step 1.6.1.2

Multiply $- 4$ by $2$ .

$y = \frac{2 \pm \sqrt{4 - 8 \cdot (8 x - 64)}}{2 \cdot 2}$

Step 1.6.1.3

Apply the distributive property.

$y = \frac{2 \pm \sqrt{4 - 8 (8 x) - 8 \cdot - 64}}{2 \cdot 2}$

Step 1.6.1.4

Multiply $8$ by $- 8$ .

$y = \frac{2 \pm \sqrt{4 - 64 x - 8 \cdot - 64}}{2 \cdot 2}$

Step 1.6.1.5

Multiply $- 8$ by $- 64$ .

$y = \frac{2 \pm \sqrt{4 - 64 x + 512}}{2 \cdot 2}$

Step 1.6.1.6

Add $4$ and $512$ .

$y = \frac{2 \pm \sqrt{- 64 x + 516}}{2 \cdot 2}$

Step 1.6.1.7

Factor $4$ out of $- 64 x + 516$ .

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Step 1.6.1.7.1

Factor $4$ out of $- 64 x$ .

$y = \frac{2 \pm \sqrt{4 (- 16 x) + 516}}{2 \cdot 2}$

Step 1.6.1.7.2

Factor $4$ out of $516$ .

$y = \frac{2 \pm \sqrt{4 (- 16 x) + 4 (129)}}{2 \cdot 2}$

Step 1.6.1.7.3

Factor $4$ out of $4 (- 16 x) + 4 (129)$ .

$y = \frac{2 \pm \sqrt{4 (- 16 x + 129)}}{2 \cdot 2}$

Step 1.6.1.8

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

$y = \frac{2 \pm \sqrt{2^{2} (- 16 x + 129)}}{2 \cdot 2}$

Step 1.6.1.9

Pull terms out from under the radical.

$y = \frac{2 \pm 2 \sqrt{- 16 x + 129}}{2 \cdot 2}$

Step 1.6.2

Multiply $2$ by $2$ .

$y = \frac{2 \pm 2 \sqrt{- 16 x + 129}}{4}$

Step 1.6.3

Simplify $\frac{2 \pm 2 \sqrt{- 16 x + 129}}{4}$ .

$y = \frac{1 \pm \sqrt{- 16 x + 129}}{2}$

Step 1.6.4

Change the $\pm$ to $+$ .

$y = \frac{1 + \sqrt{- 16 x + 129}}{2}$

Step 1.7

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

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Step 1.7.1

Simplify the numerator.

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Step 1.7.1.1

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot (8 x - 64)}}{2 \cdot 2}$

Step 1.7.1.2

Multiply $- 4$ by $2$ .

$y = \frac{2 \pm \sqrt{4 - 8 \cdot (8 x - 64)}}{2 \cdot 2}$

Step 1.7.1.3

Apply the distributive property.

$y = \frac{2 \pm \sqrt{4 - 8 (8 x) - 8 \cdot - 64}}{2 \cdot 2}$

Step 1.7.1.4

Multiply $8$ by $- 8$ .

$y = \frac{2 \pm \sqrt{4 - 64 x - 8 \cdot - 64}}{2 \cdot 2}$

Step 1.7.1.5

Multiply $- 8$ by $- 64$ .

$y = \frac{2 \pm \sqrt{4 - 64 x + 512}}{2 \cdot 2}$

Step 1.7.1.6

Add $4$ and $512$ .

$y = \frac{2 \pm \sqrt{- 64 x + 516}}{2 \cdot 2}$

Step 1.7.1.7

Factor $4$ out of $- 64 x + 516$ .

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Step 1.7.1.7.1

Factor $4$ out of $- 64 x$ .

$y = \frac{2 \pm \sqrt{4 (- 16 x) + 516}}{2 \cdot 2}$

Step 1.7.1.7.2

Factor $4$ out of $516$ .

$y = \frac{2 \pm \sqrt{4 (- 16 x) + 4 (129)}}{2 \cdot 2}$

Step 1.7.1.7.3

Factor $4$ out of $4 (- 16 x) + 4 (129)$ .

$y = \frac{2 \pm \sqrt{4 (- 16 x + 129)}}{2 \cdot 2}$

Step 1.7.1.8

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

$y = \frac{2 \pm \sqrt{2^{2} (- 16 x + 129)}}{2 \cdot 2}$

Step 1.7.1.9

Pull terms out from under the radical.

$y = \frac{2 \pm 2 \sqrt{- 16 x + 129}}{2 \cdot 2}$

Step 1.7.2

Multiply $2$ by $2$ .

$y = \frac{2 \pm 2 \sqrt{- 16 x + 129}}{4}$

Step 1.7.3

Simplify $\frac{2 \pm 2 \sqrt{- 16 x + 129}}{4}$ .

$y = \frac{1 \pm \sqrt{- 16 x + 129}}{2}$

Step 1.7.4

Change the $\pm$ to $-$ .

$y = \frac{1 - \sqrt{- 16 x + 129}}{2}$

Step 1.8

The final answer is the combination of both solutions.

$y = \frac{1 + \sqrt{- 16 x + 129}}{2}$

$y = \frac{1 - \sqrt{- 16 x + 129}}{2}$

$y = \frac{1 + \sqrt{- 16 x + 129}}{2}$

$y = \frac{1 - \sqrt{- 16 x + 129}}{2}$

Step 2

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Step 3

Split the fraction $\frac{1 + \sqrt{- 16 x + 129}}{2}$ into two fractions.

$y = \frac{1}{2} + \frac{\sqrt{- 16 x + 129}}{2}$

$y = \frac{1 - \sqrt{- 16 x + 129}}{2}$

Step 4

Split the fraction $\frac{1 - \sqrt{- 16 x + 129}}{2}$ into two fractions.

$y = \frac{1}{2} + \frac{\sqrt{- 16 x + 129}}{2}$

$y = \frac{1}{2} + \frac{- \sqrt{- 16 x + 129}}{2}$

Step 5

Move the negative in front of the fraction.

$y = \frac{1}{2} + \frac{\sqrt{- 16 x + 129}}{2}$

$y = \frac{1}{2} - \frac{\sqrt{- 16 x + 129}}{2}$

Step 6

Reorder terms.

$y = \frac{1}{2} + \frac{1}{2} \cdot \sqrt{- 16 x + 129}$

$y = \frac{1}{2} - (\frac{1}{2} \cdot \sqrt{- 16 x + 129})$

Step 7

Remove parentheses.

$y = \frac{1}{2} + \frac{1}{2} \cdot \sqrt{- 16 x + 129}$

$y = \frac{1}{2} - \frac{1}{2} \cdot \sqrt{- 16 x + 129}$

Step 8