Algebra Examples

$2 x - y^{2} = 4 y + 10$

Step 1

Solve for $y$ .

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Step 1.1

Subtract $4 y$ from both sides of the equation.

$2 x - y^{2} - 4 y = 10$

Step 1.2

Subtract $10$ from both sides of the equation.

$2 x - y^{2} - 4 y - 10 = 0$

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 $a = - 1$ , $b = - 4$ , and $c = 2 x - 10$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (- 1 \cdot (2 x - 10))}}{2 \cdot - 1}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 1 \cdot (2 x - 10)$ .

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Step 1.5.1.1.1

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

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot - 1 \cdot (2 x - 10)}}{2 \cdot - 1}$

Step 1.5.1.1.2

Factor $- 4$ out of $- 4 \cdot - 1 \cdot (2 x - 10)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (- 1 \cdot (2 x - 10))}}{2 \cdot - 1}$

Step 1.5.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 1 \cdot (2 x - 10))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - 1 \cdot (2 x - 10))}}{2 \cdot - 1}$

Step 1.5.1.2

Factor $- 1$ out of $- 4 - 1 \cdot (2 x - 10)$ .

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Step 1.5.1.2.1

Reorder $- 4$ and $- 1 \cdot (2 x - 10)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (2 x - 10) - 4)}}{2 \cdot - 1}$

Step 1.5.1.2.2

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

$y = \frac{4 \pm \sqrt{- 4 (- 1 (2 x - 10) - 1 \cdot 4)}}{2 \cdot - 1}$

Step 1.5.1.2.3

Factor $- 1$ out of $- 1 (2 x - 10) - 1 (4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (2 x - 10 + 4))}}{2 \cdot - 1}$

Step 1.5.1.2.4

Rewrite $- 1 (2 x - 10 + 4)$ as $- (2 x - 10 + 4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- (2 x - 10 + 4))}}{2 \cdot - 1}$

Step 1.5.1.3

Add $- 10$ and $4$ .

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

Step 1.5.1.4

Factor $2$ out of $2 x - 6$ .

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Step 1.5.1.4.1

Factor $2$ out of $2 x$ .

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

Step 1.5.1.4.2

Factor $2$ out of $- 6$ .

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

Step 1.5.1.4.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

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

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

Step 1.5.1.5

Multiply $- 1$ by $2$ .

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

Step 1.5.1.6

Multiply $- 4$ by $- 2$ .

$y = \frac{4 \pm \sqrt{8 (x - 3)}}{2 \cdot - 1}$

Step 1.5.1.7

Rewrite $8 (x - 3)$ as $2^{2} \cdot (2 (x - 3))$ .

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Step 1.5.1.7.1

Factor $4$ out of $8$ .

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

Step 1.5.1.7.2

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

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

Step 1.5.1.7.3

Add parentheses.

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

Step 1.5.1.8

Pull terms out from under the radical.

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

Step 1.5.2

Multiply $2$ by $- 1$ .

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

Step 1.5.3

Simplify $\frac{4 \pm 2 \sqrt{2 (x - 3)}}{- 2}$ .

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

Step 1.5.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{2 (x - 3)}}{- 1}$ .

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

Step 1.5.5

Rewrite $- 1 \cdot (2 \pm \sqrt{2 (x - 3)})$ as $- (2 \pm \sqrt{2 (x - 3)})$ .

$y = - (2 \pm \sqrt{2 (x - 3)})$

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

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 1 \cdot (2 x - 10)$ .

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Step 1.6.1.1.1

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

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot - 1 \cdot (2 x - 10)}}{2 \cdot - 1}$

Step 1.6.1.1.2

Factor $- 4$ out of $- 4 \cdot - 1 \cdot (2 x - 10)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (- 1 \cdot (2 x - 10))}}{2 \cdot - 1}$

Step 1.6.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 1 \cdot (2 x - 10))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - 1 \cdot (2 x - 10))}}{2 \cdot - 1}$

Step 1.6.1.2

Factor $- 1$ out of $- 4 - 1 \cdot (2 x - 10)$ .

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Step 1.6.1.2.1

Reorder $- 4$ and $- 1 \cdot (2 x - 10)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (2 x - 10) - 4)}}{2 \cdot - 1}$

Step 1.6.1.2.2

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

$y = \frac{4 \pm \sqrt{- 4 (- 1 (2 x - 10) - 1 \cdot 4)}}{2 \cdot - 1}$

Step 1.6.1.2.3

Factor $- 1$ out of $- 1 (2 x - 10) - 1 (4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (2 x - 10 + 4))}}{2 \cdot - 1}$

Step 1.6.1.2.4

Rewrite $- 1 (2 x - 10 + 4)$ as $- (2 x - 10 + 4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- (2 x - 10 + 4))}}{2 \cdot - 1}$

Step 1.6.1.3

Add $- 10$ and $4$ .

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

Step 1.6.1.4

Factor $2$ out of $2 x - 6$ .

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Step 1.6.1.4.1

Factor $2$ out of $2 x$ .

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

Step 1.6.1.4.2

Factor $2$ out of $- 6$ .

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

Step 1.6.1.4.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

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

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

Step 1.6.1.5

Multiply $- 1$ by $2$ .

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

Step 1.6.1.6

Multiply $- 4$ by $- 2$ .

$y = \frac{4 \pm \sqrt{8 (x - 3)}}{2 \cdot - 1}$

Step 1.6.1.7

Rewrite $8 (x - 3)$ as $2^{2} \cdot (2 (x - 3))$ .

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Step 1.6.1.7.1

Factor $4$ out of $8$ .

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

Step 1.6.1.7.2

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

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

Step 1.6.1.7.3

Add parentheses.

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

Step 1.6.1.8

Pull terms out from under the radical.

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

Step 1.6.2

Multiply $2$ by $- 1$ .

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

Step 1.6.3

Simplify $\frac{4 \pm 2 \sqrt{2 (x - 3)}}{- 2}$ .

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

Step 1.6.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{2 (x - 3)}}{- 1}$ .

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

Step 1.6.5

Rewrite $- 1 \cdot (2 \pm \sqrt{2 (x - 3)})$ as $- (2 \pm \sqrt{2 (x - 3)})$ .

$y = - (2 \pm \sqrt{2 (x - 3)})$

Step 1.6.6

Change the $\pm$ to $+$ .

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

Step 1.6.7

Apply the distributive property.

$y = - 1 \cdot 2 - \sqrt{2 (x - 3)}$

Step 1.6.8

Multiply $- 1$ by $2$ .

$y = - 2 - \sqrt{2 (x - 3)}$

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

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 1 \cdot (2 x - 10)$ .

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Step 1.7.1.1.1

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

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot - 1 \cdot (2 x - 10)}}{2 \cdot - 1}$

Step 1.7.1.1.2

Factor $- 4$ out of $- 4 \cdot - 1 \cdot (2 x - 10)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (- 1 \cdot (2 x - 10))}}{2 \cdot - 1}$

Step 1.7.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 1 \cdot (2 x - 10))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - 1 \cdot (2 x - 10))}}{2 \cdot - 1}$

Step 1.7.1.2

Factor $- 1$ out of $- 4 - 1 \cdot (2 x - 10)$ .

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Step 1.7.1.2.1

Reorder $- 4$ and $- 1 \cdot (2 x - 10)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (2 x - 10) - 4)}}{2 \cdot - 1}$

Step 1.7.1.2.2

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

$y = \frac{4 \pm \sqrt{- 4 (- 1 (2 x - 10) - 1 \cdot 4)}}{2 \cdot - 1}$

Step 1.7.1.2.3

Factor $- 1$ out of $- 1 (2 x - 10) - 1 (4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (2 x - 10 + 4))}}{2 \cdot - 1}$

Step 1.7.1.2.4

Rewrite $- 1 (2 x - 10 + 4)$ as $- (2 x - 10 + 4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- (2 x - 10 + 4))}}{2 \cdot - 1}$

Step 1.7.1.3

Add $- 10$ and $4$ .

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

Step 1.7.1.4

Factor $2$ out of $2 x - 6$ .

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Step 1.7.1.4.1

Factor $2$ out of $2 x$ .

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

Step 1.7.1.4.2

Factor $2$ out of $- 6$ .

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

Step 1.7.1.4.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

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

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

Step 1.7.1.5

Multiply $- 1$ by $2$ .

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

Step 1.7.1.6

Multiply $- 4$ by $- 2$ .

$y = \frac{4 \pm \sqrt{8 (x - 3)}}{2 \cdot - 1}$

Step 1.7.1.7

Rewrite $8 (x - 3)$ as $2^{2} \cdot (2 (x - 3))$ .

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Step 1.7.1.7.1

Factor $4$ out of $8$ .

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

Step 1.7.1.7.2

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

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

Step 1.7.1.7.3

Add parentheses.

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

Step 1.7.1.8

Pull terms out from under the radical.

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

Step 1.7.2

Multiply $2$ by $- 1$ .

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

Step 1.7.3

Simplify $\frac{4 \pm 2 \sqrt{2 (x - 3)}}{- 2}$ .

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

Step 1.7.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{2 (x - 3)}}{- 1}$ .

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

Step 1.7.5

Rewrite $- 1 \cdot (2 \pm \sqrt{2 (x - 3)})$ as $- (2 \pm \sqrt{2 (x - 3)})$ .

$y = - (2 \pm \sqrt{2 (x - 3)})$

Step 1.7.6

Change the $\pm$ to $-$ .

$y = - (2 - \sqrt{2 (x - 3)})$

Step 1.7.7

Apply the distributive property.

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

Step 1.7.8

Multiply $- 1$ by $2$ .

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

Step 1.7.9

Multiply $- - \sqrt{2 (x - 3)}$ .

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Step 1.7.9.1

Multiply $- 1$ by $- 1$ .

$y = - 2 + 1 \sqrt{2 (x - 3)}$

Step 1.7.9.2

Multiply $\sqrt{2 (x - 3)}$ by $1$ .

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

Step 1.8

The final answer is the combination of both solutions.

$y = - 2 - \sqrt{2 (x - 3)}$

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

$y = - 2 - \sqrt{2 (x - 3)}$

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

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

The standard form is $- 2 - \sqrt{2 (x - 3)}, - 2 + \sqrt{2 (x - 3)}$ .

$y = - 2 - \sqrt{2 (x - 3)}$

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

Step 4