Algebra Examples

$y + 2 x^{2} = 5 - 2 x$

Step 1

Add $2 x$ to both sides of the equation.

$y + 2 x^{2} + 2 x = 5$

Step 2

Subtract $5$ from both sides of the equation.

$y + 2 x^{2} + 2 x - 5 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 2$ , $b = 2$ , and $c = y - 5$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (2 \cdot (y - 5))}}{2 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor $2$ out of $2^{2} - 4 \cdot 2 \cdot (y - 5)$ .

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Step 5.1.1.1

Factor $2$ out of $2^{2}$ .

$x = \frac{- 2 \pm \sqrt{2 \cdot 2 - 4 \cdot 2 \cdot (y - 5)}}{2 \cdot 2}$

Step 5.1.1.2

Factor $2$ out of $- 4 \cdot 2 \cdot (y - 5)$ .

$x = \frac{- 2 \pm \sqrt{2 \cdot 2 + 2 (- 2 \cdot 2 \cdot (y - 5))}}{2 \cdot 2}$

Step 5.1.1.3

Factor $2$ out of $2 \cdot 2 + 2 (- 2 \cdot 2 \cdot (y - 5))$ .

$x = \frac{- 2 \pm \sqrt{2 (2 - 2 \cdot 2 \cdot (y - 5))}}{2 \cdot 2}$

Step 5.1.2

Factor $2$ out of $2 - 2 \cdot 2 \cdot (y - 5)$ .

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Step 5.1.2.1

Factor $2$ out of $2$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1) - 2 \cdot 2 \cdot (y - 5))}}{2 \cdot 2}$

Step 5.1.2.2

Factor $2$ out of $- 2 \cdot 2 \cdot (y - 5)$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1) + 2 (- 1 \cdot 2 \cdot (y - 5)))}}{2 \cdot 2}$

Step 5.1.2.3

Factor $2$ out of $2 (1) + 2 (- 1 \cdot 2 \cdot (y - 5))$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1 - 1 \cdot 2 \cdot (y - 5)))}}{2 \cdot 2}$

Step 5.1.3

Multiply $- 1$ by $2$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1 - 2 \cdot (y - 5)))}}{2 \cdot 2}$

Step 5.1.4

Apply the distributive property.

$x = \frac{- 2 \pm \sqrt{2 (2 (1 - 2 y - 2 \cdot - 5))}}{2 \cdot 2}$

Step 5.1.5

Multiply $- 2$ by $- 5$ .

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

Step 5.1.6

Add $1$ and $10$ .

$x = \frac{- 2 \pm \sqrt{2 \cdot (2 (- 2 y + 11))}}{2 \cdot 2}$

Step 5.1.7

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 (- 2 y + 11)}}{2 \cdot 2}$

Step 5.1.8

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

$x = \frac{- 2 \pm \sqrt{2^{2} (- 2 y + 11)}}{2 \cdot 2}$

Step 5.1.9

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{- 2 y + 11}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm 2 \sqrt{- 2 y + 11}}{4}$

Step 5.3

Simplify $\frac{- 2 \pm 2 \sqrt{- 2 y + 11}}{4}$ .

$x = \frac{- 1 \pm \sqrt{- 2 y + 11}}{2}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Factor $2$ out of $2^{2} - 4 \cdot 2 \cdot (y - 5)$ .

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Step 6.1.1.1

Factor $2$ out of $2^{2}$ .

$x = \frac{- 2 \pm \sqrt{2 \cdot 2 - 4 \cdot 2 \cdot (y - 5)}}{2 \cdot 2}$

Step 6.1.1.2

Factor $2$ out of $- 4 \cdot 2 \cdot (y - 5)$ .

$x = \frac{- 2 \pm \sqrt{2 \cdot 2 + 2 (- 2 \cdot 2 \cdot (y - 5))}}{2 \cdot 2}$

Step 6.1.1.3

Factor $2$ out of $2 \cdot 2 + 2 (- 2 \cdot 2 \cdot (y - 5))$ .

$x = \frac{- 2 \pm \sqrt{2 (2 - 2 \cdot 2 \cdot (y - 5))}}{2 \cdot 2}$

Step 6.1.2

Factor $2$ out of $2 - 2 \cdot 2 \cdot (y - 5)$ .

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Step 6.1.2.1

Factor $2$ out of $2$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1) - 2 \cdot 2 \cdot (y - 5))}}{2 \cdot 2}$

Step 6.1.2.2

Factor $2$ out of $- 2 \cdot 2 \cdot (y - 5)$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1) + 2 (- 1 \cdot 2 \cdot (y - 5)))}}{2 \cdot 2}$

Step 6.1.2.3

Factor $2$ out of $2 (1) + 2 (- 1 \cdot 2 \cdot (y - 5))$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1 - 1 \cdot 2 \cdot (y - 5)))}}{2 \cdot 2}$

Step 6.1.3

Multiply $- 1$ by $2$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1 - 2 \cdot (y - 5)))}}{2 \cdot 2}$

Step 6.1.4

Apply the distributive property.

$x = \frac{- 2 \pm \sqrt{2 (2 (1 - 2 y - 2 \cdot - 5))}}{2 \cdot 2}$

Step 6.1.5

Multiply $- 2$ by $- 5$ .

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

Step 6.1.6

Add $1$ and $10$ .

$x = \frac{- 2 \pm \sqrt{2 \cdot (2 (- 2 y + 11))}}{2 \cdot 2}$

Step 6.1.7

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 (- 2 y + 11)}}{2 \cdot 2}$

Step 6.1.8

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

$x = \frac{- 2 \pm \sqrt{2^{2} (- 2 y + 11)}}{2 \cdot 2}$

Step 6.1.9

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{- 2 y + 11}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm 2 \sqrt{- 2 y + 11}}{4}$

Step 6.3

Simplify $\frac{- 2 \pm 2 \sqrt{- 2 y + 11}}{4}$ .

$x = \frac{- 1 \pm \sqrt{- 2 y + 11}}{2}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{- 2 y + 11}}{2}$

Step 6.5

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

$x = \frac{- 1 \cdot 1 + \sqrt{- 2 y + 11}}{2}$

Step 6.6

Factor $- 1$ out of $\sqrt{- 2 y + 11}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{- 2 y + 11})}{2}$

Step 6.7

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{- 2 y + 11})$ .

$x = \frac{- 1 (1 - \sqrt{- 2 y + 11})}{2}$

Step 6.8

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{- 2 y + 11}}{2}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Factor $2$ out of $2^{2} - 4 \cdot 2 \cdot (y - 5)$ .

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Step 7.1.1.1

Factor $2$ out of $2^{2}$ .

$x = \frac{- 2 \pm \sqrt{2 \cdot 2 - 4 \cdot 2 \cdot (y - 5)}}{2 \cdot 2}$

Step 7.1.1.2

Factor $2$ out of $- 4 \cdot 2 \cdot (y - 5)$ .

$x = \frac{- 2 \pm \sqrt{2 \cdot 2 + 2 (- 2 \cdot 2 \cdot (y - 5))}}{2 \cdot 2}$

Step 7.1.1.3

Factor $2$ out of $2 \cdot 2 + 2 (- 2 \cdot 2 \cdot (y - 5))$ .

$x = \frac{- 2 \pm \sqrt{2 (2 - 2 \cdot 2 \cdot (y - 5))}}{2 \cdot 2}$

Step 7.1.2

Factor $2$ out of $2 - 2 \cdot 2 \cdot (y - 5)$ .

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Step 7.1.2.1

Factor $2$ out of $2$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1) - 2 \cdot 2 \cdot (y - 5))}}{2 \cdot 2}$

Step 7.1.2.2

Factor $2$ out of $- 2 \cdot 2 \cdot (y - 5)$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1) + 2 (- 1 \cdot 2 \cdot (y - 5)))}}{2 \cdot 2}$

Step 7.1.2.3

Factor $2$ out of $2 (1) + 2 (- 1 \cdot 2 \cdot (y - 5))$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1 - 1 \cdot 2 \cdot (y - 5)))}}{2 \cdot 2}$

Step 7.1.3

Multiply $- 1$ by $2$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1 - 2 \cdot (y - 5)))}}{2 \cdot 2}$

Step 7.1.4

Apply the distributive property.

$x = \frac{- 2 \pm \sqrt{2 (2 (1 - 2 y - 2 \cdot - 5))}}{2 \cdot 2}$

Step 7.1.5

Multiply $- 2$ by $- 5$ .

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

Step 7.1.6

Add $1$ and $10$ .

$x = \frac{- 2 \pm \sqrt{2 \cdot (2 (- 2 y + 11))}}{2 \cdot 2}$

Step 7.1.7

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 (- 2 y + 11)}}{2 \cdot 2}$

Step 7.1.8

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

$x = \frac{- 2 \pm \sqrt{2^{2} (- 2 y + 11)}}{2 \cdot 2}$

Step 7.1.9

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{- 2 y + 11}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm 2 \sqrt{- 2 y + 11}}{4}$

Step 7.3

Simplify $\frac{- 2 \pm 2 \sqrt{- 2 y + 11}}{4}$ .

$x = \frac{- 1 \pm \sqrt{- 2 y + 11}}{2}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{- 2 y + 11}}{2}$

Step 7.5

Factor $- 1$ out of $- 1 - \sqrt{- 2 y + 11}$ .

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Step 7.5.1

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

$x = \frac{- 1 \cdot 1 - \sqrt{- 2 y + 11}}{2}$

Step 7.5.2

Factor $- 1$ out of $- \sqrt{- 2 y + 11}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{- 2 y + 11})}{2}$

Step 7.5.3

Factor $- 1$ out of $- 1 (1) - (\sqrt{- 2 y + 11})$ .

$x = \frac{- 1 (1 + \sqrt{- 2 y + 11})}{2}$

Step 7.5.4

Rewrite $- 1 (1 + \sqrt{- 2 y + 11})$ as $- (1 + \sqrt{- 2 y + 11})$ .

$x = \frac{- (1 + \sqrt{- 2 y + 11})}{2}$

Step 7.6

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{- 2 y + 11}}{2}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{- 2 y + 11}}{2}$

$x = - \frac{1 + \sqrt{- 2 y + 11}}{2}$