Algebra Examples

$8 x + 3 y + y^{2} = - 263 - x^{2}$

Step 1

Solve for $y$ .

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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

Add $263$ to both sides of the equation.

$8 x + 3 y + y^{2} + 263 = - x^{2}$

Step 1.1.2

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

$8 x + 3 y + y^{2} + 263 + x^{2} = 0$

Step 1.2

Use the quadratic formula to find the solutions.

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

Step 1.3

Substitute the values $a = 1$ , $b = 3$ , and $c = 8 x + 263 + x^{2}$ into the quadratic formula and solve for $y$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot (8 x + 263 + x^{2}))}}{2 \cdot 1}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Raise $3$ to the power of $2$ .

$y = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot (8 x + 263 + x^{2})}}{2 \cdot 1}$

Step 1.4.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 3 \pm \sqrt{9 - 4 \cdot (8 x + 263 + x^{2})}}{2 \cdot 1}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{- 3 \pm \sqrt{9 - 4 (8 x) - 4 \cdot 263 - 4 x^{2}}}{2 \cdot 1}$

Step 1.4.1.4

Simplify.

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Step 1.4.1.4.1

Multiply $8$ by $- 4$ .

$y = \frac{- 3 \pm \sqrt{9 - 32 x - 4 \cdot 263 - 4 x^{2}}}{2 \cdot 1}$

Step 1.4.1.4.2

Multiply $- 4$ by $263$ .

$y = \frac{- 3 \pm \sqrt{9 - 32 x - 1052 - 4 x^{2}}}{2 \cdot 1}$

Step 1.4.1.5

Subtract $1052$ from $9$ .

$y = \frac{- 3 \pm \sqrt{- 32 x - 1043 - 4 x^{2}}}{2 \cdot 1}$

Step 1.4.1.6

Reorder terms.

$y = \frac{- 3 \pm \sqrt{- 4 x^{2} - 32 x - 1043}}{2 \cdot 1}$

Step 1.4.2

Multiply $2$ by $1$ .

$y = \frac{- 3 \pm \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

Step 1.5

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

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Raise $3$ to the power of $2$ .

$y = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot (8 x + 263 + x^{2})}}{2 \cdot 1}$

Step 1.5.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 3 \pm \sqrt{9 - 4 \cdot (8 x + 263 + x^{2})}}{2 \cdot 1}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{- 3 \pm \sqrt{9 - 4 (8 x) - 4 \cdot 263 - 4 x^{2}}}{2 \cdot 1}$

Step 1.5.1.4

Simplify.

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Step 1.5.1.4.1

Multiply $8$ by $- 4$ .

$y = \frac{- 3 \pm \sqrt{9 - 32 x - 4 \cdot 263 - 4 x^{2}}}{2 \cdot 1}$

Step 1.5.1.4.2

Multiply $- 4$ by $263$ .

$y = \frac{- 3 \pm \sqrt{9 - 32 x - 1052 - 4 x^{2}}}{2 \cdot 1}$

Step 1.5.1.5

Subtract $1052$ from $9$ .

$y = \frac{- 3 \pm \sqrt{- 32 x - 1043 - 4 x^{2}}}{2 \cdot 1}$

Step 1.5.1.6

Reorder terms.

$y = \frac{- 3 \pm \sqrt{- 4 x^{2} - 32 x - 1043}}{2 \cdot 1}$

Step 1.5.2

Multiply $2$ by $1$ .

$y = \frac{- 3 \pm \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

Step 1.5.3

Change the $\pm$ to $+$ .

$y = \frac{- 3 + \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

Step 1.5.4

Rewrite $- 3$ as $- 1 (3)$ .

$y = \frac{- 1 \cdot 3 + \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

Step 1.5.5

Factor $- 1$ out of $\sqrt{- 4 x^{2} - 32 x - 1043}$ .

$y = \frac{- 1 \cdot 3 - 1 (- \sqrt{- 4 x^{2} - 32 x - 1043})}{2}$

Step 1.5.6

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{- 4 x^{2} - 32 x - 1043})$ .

$y = \frac{- 1 (3 - \sqrt{- 4 x^{2} - 32 x - 1043})}{2}$

Step 1.5.7

Move the negative in front of the fraction.

$y = - \frac{3 - \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

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 $3$ to the power of $2$ .

$y = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot (8 x + 263 + x^{2})}}{2 \cdot 1}$

Step 1.6.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 3 \pm \sqrt{9 - 4 \cdot (8 x + 263 + x^{2})}}{2 \cdot 1}$

Step 1.6.1.3

Apply the distributive property.

$y = \frac{- 3 \pm \sqrt{9 - 4 (8 x) - 4 \cdot 263 - 4 x^{2}}}{2 \cdot 1}$

Step 1.6.1.4

Simplify.

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Step 1.6.1.4.1

Multiply $8$ by $- 4$ .

$y = \frac{- 3 \pm \sqrt{9 - 32 x - 4 \cdot 263 - 4 x^{2}}}{2 \cdot 1}$

Step 1.6.1.4.2

Multiply $- 4$ by $263$ .

$y = \frac{- 3 \pm \sqrt{9 - 32 x - 1052 - 4 x^{2}}}{2 \cdot 1}$

Step 1.6.1.5

Subtract $1052$ from $9$ .

$y = \frac{- 3 \pm \sqrt{- 32 x - 1043 - 4 x^{2}}}{2 \cdot 1}$

Step 1.6.1.6

Reorder terms.

$y = \frac{- 3 \pm \sqrt{- 4 x^{2} - 32 x - 1043}}{2 \cdot 1}$

Step 1.6.2

Multiply $2$ by $1$ .

$y = \frac{- 3 \pm \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

Step 1.6.3

Change the $\pm$ to $-$ .

$y = \frac{- 3 - \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

Step 1.6.4

Factor $- 1$ out of $- 3 - \sqrt{- 4 x^{2} - 32 x - 1043}$ .

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Step 1.6.4.1

Rewrite $- 3$ as $- 1 (3)$ .

$y = \frac{- 1 \cdot 3 - \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

Step 1.6.4.2

Factor $- 1$ out of $- \sqrt{- 4 x^{2} - 32 x - 1043}$ .

$y = \frac{- 1 \cdot 3 - (\sqrt{- 4 x^{2} - 32 x - 1043})}{2}$

Step 1.6.4.3

Factor $- 1$ out of $- 1 (3) - (\sqrt{- 4 x^{2} - 32 x - 1043})$ .

$y = \frac{- 1 (3 + \sqrt{- 4 x^{2} - 32 x - 1043})}{2}$

Step 1.6.4.4

Rewrite $- 1 (3 + \sqrt{- 4 x^{2} - 32 x - 1043})$ as $- (3 + \sqrt{- 4 x^{2} - 32 x - 1043})$ .

$y = \frac{- (3 + \sqrt{- 4 x^{2} - 32 x - 1043})}{2}$

Step 1.6.5

Move the negative in front of the fraction.

$y = - \frac{3 + \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

Step 1.7

The final answer is the combination of both solutions.

$y = - \frac{3 - \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

$y = - \frac{3 + \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

$y = - \frac{3 - \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

$y = - \frac{3 + \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

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 $\frac{3 - \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$ into two fractions.

$y = - (\frac{3}{2} + \frac{- \sqrt{- 4 x^{2} - 32 x - 1043}}{2})$

$y = - \frac{3 + \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

Step 4

Move the negative in front of the fraction.

$y = - (\frac{3}{2} - \frac{\sqrt{- 4 x^{2} - 32 x - 1043}}{2})$

$y = - \frac{3 + \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

Step 5

Apply the distributive property.

$y = - \frac{3}{2} + \frac{\sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

$y = - \frac{3 + \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

Step 6

Split the fraction $\frac{3 + \sqrt{- 4 x^{2} - 32 x - 1043}}{2}$ into two fractions.

$y = - \frac{3}{2} + \frac{\sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

$y = - (\frac{3}{2} + \frac{\sqrt{- 4 x^{2} - 32 x - 1043}}{2})$

Step 7

Apply the distributive property.

$y = - \frac{3}{2} + \frac{\sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

$y = - \frac{3}{2} - \frac{\sqrt{- 4 x^{2} - 32 x - 1043}}{2}$

Step 8

Reorder terms.

$y = - \frac{3}{2} + \frac{1}{2} \cdot \sqrt{- 4 x^{2} - 32 x - 1043}$

$y = - \frac{3}{2} - (\frac{1}{2} \cdot \sqrt{- 4 x^{2} - 32 x - 1043})$

Step 9

Remove parentheses.

$y = - \frac{3}{2} + \frac{1}{2} \cdot \sqrt{- 4 x^{2} - 32 x - 1043}$

$y = - \frac{3}{2} - \frac{1}{2} \cdot \sqrt{- 4 x^{2} - 32 x - 1043}$

Step 10