Algebra Examples

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

Step 1

Solve for $y$ .

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Step 1.1

Rewrite the equation as $\frac{1}{2} y^{2} + 2 y + 11 = x$ .

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

Step 1.2

Combine $\frac{1}{2}$ and $y^{2}$ .

$\frac{y^{2}}{2} + 2 y + 11 = x$

Step 1.3

Subtract $x$ from both sides of the equation.

$\frac{y^{2}}{2} + 2 y + 11 - x = 0$

Step 1.4

Multiply through by the least common denominator $2$ , then simplify.

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Step 1.4.1

Apply the distributive property.

$2 (\frac{y^{2}}{2}) + 2 (2 y) + 2 \cdot 11 + 2 (- x) = 0$

Step 1.4.2

Simplify.

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Step 1.4.2.1

Cancel the common factor of $2$ .

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Step 1.4.2.1.1

Cancel the common factor.

$2 (\frac{y^{2}}{2}) + 2 (2 y) + 2 \cdot 11 + 2 (- x) = 0$

Step 1.4.2.1.2

Rewrite the expression.

$y^{2} + 2 (2 y) + 2 \cdot 11 + 2 (- x) = 0$

Step 1.4.2.2

Multiply $2$ by $2$ .

$y^{2} + 4 y + 2 \cdot 11 + 2 (- x) = 0$

Step 1.4.2.3

Multiply $2$ by $11$ .

$y^{2} + 4 y + 22 + 2 (- x) = 0$

Step 1.4.2.4

Multiply $- 1$ by $2$ .

$y^{2} + 4 y + 22 - 2 x = 0$

Step 1.4.3

Move $22$ .

$y^{2} + 4 y - 2 x + 22 = 0$

Step 1.4.4

Move $4 y$ .

$y^{2} - 2 x + 4 y + 22 = 0$

Step 1.5

Use the quadratic formula to find the solutions.

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

Step 1.6

Substitute the values $a = 1$ , $b = 4$ , and $c = - 2 x + 22$ into the quadratic formula and solve for $y$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot (- 2 x + 22))}}{2 \cdot 1}$

Step 1.7

Simplify.

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Step 1.7.1

Simplify the numerator.

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Step 1.7.1.1

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

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (- 2 x + 22)}}{2 \cdot 1}$

Step 1.7.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot (- 2 x + 22)}}{2 \cdot 1}$

Step 1.7.1.3

Apply the distributive property.

$y = \frac{- 4 \pm \sqrt{16 - 4 (- 2 x) - 4 \cdot 22}}{2 \cdot 1}$

Step 1.7.1.4

Multiply $- 2$ by $- 4$ .

$y = \frac{- 4 \pm \sqrt{16 + 8 x - 4 \cdot 22}}{2 \cdot 1}$

Step 1.7.1.5

Multiply $- 4$ by $22$ .

$y = \frac{- 4 \pm \sqrt{16 + 8 x - 88}}{2 \cdot 1}$

Step 1.7.1.6

Subtract $88$ from $16$ .

$y = \frac{- 4 \pm \sqrt{8 x - 72}}{2 \cdot 1}$

Step 1.7.1.7

Factor $8$ out of $8 x - 72$ .

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Step 1.7.1.7.1

Factor $8$ out of $8 x$ .

$y = \frac{- 4 \pm \sqrt{8 (x) - 72}}{2 \cdot 1}$

Step 1.7.1.7.2

Factor $8$ out of $- 72$ .

$y = \frac{- 4 \pm \sqrt{8 x + 8 \cdot - 9}}{2 \cdot 1}$

Step 1.7.1.7.3

Factor $8$ out of $8 x + 8 \cdot - 9$ .

$y = \frac{- 4 \pm \sqrt{8 (x - 9)}}{2 \cdot 1}$

Step 1.7.1.8

Rewrite $8 (x - 9)$ as $2^{2} \cdot (2 (x - 9))$ .

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Step 1.7.1.8.1

Factor $4$ out of $8$ .

$y = \frac{- 4 \pm \sqrt{4 (2) (x - 9)}}{2 \cdot 1}$

Step 1.7.1.8.2

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

$y = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (x - 9))}}{2 \cdot 1}$

Step 1.7.1.8.3

Add parentheses.

$y = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (x - 9))}}{2 \cdot 1}$

Step 1.7.1.9

Pull terms out from under the radical.

$y = \frac{- 4 \pm 2 \sqrt{2 (x - 9)}}{2 \cdot 1}$

Step 1.7.2

Multiply $2$ by $1$ .

$y = \frac{- 4 \pm 2 \sqrt{2 (x - 9)}}{2}$

Step 1.7.3

Simplify $\frac{- 4 \pm 2 \sqrt{2 (x - 9)}}{2}$ .

$y = - 2 \pm \sqrt{2 (x - 9)}$

Step 1.8

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

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Step 1.8.1

Simplify the numerator.

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Step 1.8.1.1

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

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (- 2 x + 22)}}{2 \cdot 1}$

Step 1.8.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot (- 2 x + 22)}}{2 \cdot 1}$

Step 1.8.1.3

Apply the distributive property.

$y = \frac{- 4 \pm \sqrt{16 - 4 (- 2 x) - 4 \cdot 22}}{2 \cdot 1}$

Step 1.8.1.4

Multiply $- 2$ by $- 4$ .

$y = \frac{- 4 \pm \sqrt{16 + 8 x - 4 \cdot 22}}{2 \cdot 1}$

Step 1.8.1.5

Multiply $- 4$ by $22$ .

$y = \frac{- 4 \pm \sqrt{16 + 8 x - 88}}{2 \cdot 1}$

Step 1.8.1.6

Subtract $88$ from $16$ .

$y = \frac{- 4 \pm \sqrt{8 x - 72}}{2 \cdot 1}$

Step 1.8.1.7

Factor $8$ out of $8 x - 72$ .

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Step 1.8.1.7.1

Factor $8$ out of $8 x$ .

$y = \frac{- 4 \pm \sqrt{8 (x) - 72}}{2 \cdot 1}$

Step 1.8.1.7.2

Factor $8$ out of $- 72$ .

$y = \frac{- 4 \pm \sqrt{8 x + 8 \cdot - 9}}{2 \cdot 1}$

Step 1.8.1.7.3

Factor $8$ out of $8 x + 8 \cdot - 9$ .

$y = \frac{- 4 \pm \sqrt{8 (x - 9)}}{2 \cdot 1}$

Step 1.8.1.8

Rewrite $8 (x - 9)$ as $2^{2} \cdot (2 (x - 9))$ .

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Step 1.8.1.8.1

Factor $4$ out of $8$ .

$y = \frac{- 4 \pm \sqrt{4 (2) (x - 9)}}{2 \cdot 1}$

Step 1.8.1.8.2

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

$y = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (x - 9))}}{2 \cdot 1}$

Step 1.8.1.8.3

Add parentheses.

$y = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (x - 9))}}{2 \cdot 1}$

Step 1.8.1.9

Pull terms out from under the radical.

$y = \frac{- 4 \pm 2 \sqrt{2 (x - 9)}}{2 \cdot 1}$

Step 1.8.2

Multiply $2$ by $1$ .

$y = \frac{- 4 \pm 2 \sqrt{2 (x - 9)}}{2}$

Step 1.8.3

Simplify $\frac{- 4 \pm 2 \sqrt{2 (x - 9)}}{2}$ .

$y = - 2 \pm \sqrt{2 (x - 9)}$

Step 1.8.4

Change the $\pm$ to $+$ .

$y = - 2 + \sqrt{2 (x - 9)}$

Step 1.9

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

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Step 1.9.1

Simplify the numerator.

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Step 1.9.1.1

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

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (- 2 x + 22)}}{2 \cdot 1}$

Step 1.9.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 4 \pm \sqrt{16 - 4 \cdot (- 2 x + 22)}}{2 \cdot 1}$

Step 1.9.1.3

Apply the distributive property.

$y = \frac{- 4 \pm \sqrt{16 - 4 (- 2 x) - 4 \cdot 22}}{2 \cdot 1}$

Step 1.9.1.4

Multiply $- 2$ by $- 4$ .

$y = \frac{- 4 \pm \sqrt{16 + 8 x - 4 \cdot 22}}{2 \cdot 1}$

Step 1.9.1.5

Multiply $- 4$ by $22$ .

$y = \frac{- 4 \pm \sqrt{16 + 8 x - 88}}{2 \cdot 1}$

Step 1.9.1.6

Subtract $88$ from $16$ .

$y = \frac{- 4 \pm \sqrt{8 x - 72}}{2 \cdot 1}$

Step 1.9.1.7

Factor $8$ out of $8 x - 72$ .

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Step 1.9.1.7.1

Factor $8$ out of $8 x$ .

$y = \frac{- 4 \pm \sqrt{8 (x) - 72}}{2 \cdot 1}$

Step 1.9.1.7.2

Factor $8$ out of $- 72$ .

$y = \frac{- 4 \pm \sqrt{8 x + 8 \cdot - 9}}{2 \cdot 1}$

Step 1.9.1.7.3

Factor $8$ out of $8 x + 8 \cdot - 9$ .

$y = \frac{- 4 \pm \sqrt{8 (x - 9)}}{2 \cdot 1}$

Step 1.9.1.8

Rewrite $8 (x - 9)$ as $2^{2} \cdot (2 (x - 9))$ .

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Step 1.9.1.8.1

Factor $4$ out of $8$ .

$y = \frac{- 4 \pm \sqrt{4 (2) (x - 9)}}{2 \cdot 1}$

Step 1.9.1.8.2

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

$y = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (x - 9))}}{2 \cdot 1}$

Step 1.9.1.8.3

Add parentheses.

$y = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (x - 9))}}{2 \cdot 1}$

Step 1.9.1.9

Pull terms out from under the radical.

$y = \frac{- 4 \pm 2 \sqrt{2 (x - 9)}}{2 \cdot 1}$

Step 1.9.2

Multiply $2$ by $1$ .

$y = \frac{- 4 \pm 2 \sqrt{2 (x - 9)}}{2}$

Step 1.9.3

Simplify $\frac{- 4 \pm 2 \sqrt{2 (x - 9)}}{2}$ .

$y = - 2 \pm \sqrt{2 (x - 9)}$

Step 1.9.4

Change the $\pm$ to $-$ .

$y = - 2 - \sqrt{2 (x - 9)}$

Step 1.10

The final answer is the combination of both solutions.

$y = - 2 + \sqrt{2 (x - 9)}$

$y = - 2 - \sqrt{2 (x - 9)}$

$y = - 2 + \sqrt{2 (x - 9)}$

$y = - 2 - \sqrt{2 (x - 9)}$

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 $- 2 + \sqrt{2 (x - 9)}, - 2 - \sqrt{2 (x - 9)}$ .

$y = - 2 + \sqrt{2 (x - 9)}$

$y = - 2 - \sqrt{2 (x - 9)}$

Step 4