Algebra Examples

$x + 3 y^{2} + 12 y = 18$

Step 1

Solve for $y$ .

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Step 1.1

Subtract $18$ from both sides of the equation.

$x + 3 y^{2} + 12 y - 18 = 0$

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 $a = 3$ , $b = 12$ , and $c = x - 18$ into the quadratic formula and solve for $y$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (3 \cdot (x - 18))}}{2 \cdot 3}$

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 $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot 3 \cdot (x - 18)}}{2 \cdot 3}$

Step 1.4.1.2

Multiply $- 4$ by $3$ .

$y = \frac{- 12 \pm \sqrt{144 - 12 \cdot (x - 18)}}{2 \cdot 3}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 12 x - 12 \cdot - 18}}{2 \cdot 3}$

Step 1.4.1.4

Multiply $- 12$ by $- 18$ .

$y = \frac{- 12 \pm \sqrt{144 - 12 x + 216}}{2 \cdot 3}$

Step 1.4.1.5

Add $144$ and $216$ .

$y = \frac{- 12 \pm \sqrt{- 12 x + 360}}{2 \cdot 3}$

Step 1.4.1.6

Factor $12$ out of $- 12 x + 360$ .

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Step 1.4.1.6.1

Factor $12$ out of $- 12 x$ .

$y = \frac{- 12 \pm \sqrt{12 (- x) + 360}}{2 \cdot 3}$

Step 1.4.1.6.2

Factor $12$ out of $360$ .

$y = \frac{- 12 \pm \sqrt{12 (- x) + 12 (30)}}{2 \cdot 3}$

Step 1.4.1.6.3

Factor $12$ out of $12 (- x) + 12 (30)$ .

$y = \frac{- 12 \pm \sqrt{12 (- x + 30)}}{2 \cdot 3}$

Step 1.4.1.7

Rewrite $12 (- x + 30)$ as $2^{2} \cdot (3 (- x + 30))$ .

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Step 1.4.1.7.1

Factor $4$ out of $12$ .

$y = \frac{- 12 \pm \sqrt{4 (3) (- x + 30)}}{2 \cdot 3}$

Step 1.4.1.7.2

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

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (3 (- x + 30))}}{2 \cdot 3}$

Step 1.4.1.7.3

Add parentheses.

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (3 (- x + 30))}}{2 \cdot 3}$

Step 1.4.1.8

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{3 (- x + 30)}}{2 \cdot 3}$

Step 1.4.2

Multiply $2$ by $3$ .

$y = \frac{- 12 \pm 2 \sqrt{3 (- x + 30)}}{6}$

Step 1.4.3

Simplify $\frac{- 12 \pm 2 \sqrt{3 (- x + 30)}}{6}$ .

$y = \frac{- 6 \pm \sqrt{3 (- x + 30)}}{3}$

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 $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot 3 \cdot (x - 18)}}{2 \cdot 3}$

Step 1.5.1.2

Multiply $- 4$ by $3$ .

$y = \frac{- 12 \pm \sqrt{144 - 12 \cdot (x - 18)}}{2 \cdot 3}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 12 x - 12 \cdot - 18}}{2 \cdot 3}$

Step 1.5.1.4

Multiply $- 12$ by $- 18$ .

$y = \frac{- 12 \pm \sqrt{144 - 12 x + 216}}{2 \cdot 3}$

Step 1.5.1.5

Add $144$ and $216$ .

$y = \frac{- 12 \pm \sqrt{- 12 x + 360}}{2 \cdot 3}$

Step 1.5.1.6

Factor $12$ out of $- 12 x + 360$ .

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Step 1.5.1.6.1

Factor $12$ out of $- 12 x$ .

$y = \frac{- 12 \pm \sqrt{12 (- x) + 360}}{2 \cdot 3}$

Step 1.5.1.6.2

Factor $12$ out of $360$ .

$y = \frac{- 12 \pm \sqrt{12 (- x) + 12 (30)}}{2 \cdot 3}$

Step 1.5.1.6.3

Factor $12$ out of $12 (- x) + 12 (30)$ .

$y = \frac{- 12 \pm \sqrt{12 (- x + 30)}}{2 \cdot 3}$

Step 1.5.1.7

Rewrite $12 (- x + 30)$ as $2^{2} \cdot (3 (- x + 30))$ .

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Step 1.5.1.7.1

Factor $4$ out of $12$ .

$y = \frac{- 12 \pm \sqrt{4 (3) (- x + 30)}}{2 \cdot 3}$

Step 1.5.1.7.2

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

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (3 (- x + 30))}}{2 \cdot 3}$

Step 1.5.1.7.3

Add parentheses.

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (3 (- x + 30))}}{2 \cdot 3}$

Step 1.5.1.8

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{3 (- x + 30)}}{2 \cdot 3}$

Step 1.5.2

Multiply $2$ by $3$ .

$y = \frac{- 12 \pm 2 \sqrt{3 (- x + 30)}}{6}$

Step 1.5.3

Simplify $\frac{- 12 \pm 2 \sqrt{3 (- x + 30)}}{6}$ .

$y = \frac{- 6 \pm \sqrt{3 (- x + 30)}}{3}$

Step 1.5.4

Change the $\pm$ to $+$ .

$y = \frac{- 6 + \sqrt{3 (- x + 30)}}{3}$

Step 1.5.5

Rewrite $- 6$ as $- 1 (6)$ .

$y = \frac{- 1 \cdot 6 + \sqrt{3 (- x + 30)}}{3}$

Step 1.5.6

Factor $- 1$ out of $\sqrt{3 (- x + 30)}$ .

$y = \frac{- 1 \cdot 6 - 1 (- \sqrt{3 (- x + 30)})}{3}$

Step 1.5.7

Factor $- 1$ out of $- 1 (6) - 1 (- \sqrt{3 (- x + 30)})$ .

$y = \frac{- 1 (6 - \sqrt{3 (- x + 30)})}{3}$

Step 1.5.8

Move the negative in front of the fraction.

$y = - \frac{6 - \sqrt{3 (- x + 30)}}{3}$

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

Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot 3 \cdot (x - 18)}}{2 \cdot 3}$

Step 1.6.1.2

Multiply $- 4$ by $3$ .

$y = \frac{- 12 \pm \sqrt{144 - 12 \cdot (x - 18)}}{2 \cdot 3}$

Step 1.6.1.3

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 12 x - 12 \cdot - 18}}{2 \cdot 3}$

Step 1.6.1.4

Multiply $- 12$ by $- 18$ .

$y = \frac{- 12 \pm \sqrt{144 - 12 x + 216}}{2 \cdot 3}$

Step 1.6.1.5

Add $144$ and $216$ .

$y = \frac{- 12 \pm \sqrt{- 12 x + 360}}{2 \cdot 3}$

Step 1.6.1.6

Factor $12$ out of $- 12 x + 360$ .

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Step 1.6.1.6.1

Factor $12$ out of $- 12 x$ .

$y = \frac{- 12 \pm \sqrt{12 (- x) + 360}}{2 \cdot 3}$

Step 1.6.1.6.2

Factor $12$ out of $360$ .

$y = \frac{- 12 \pm \sqrt{12 (- x) + 12 (30)}}{2 \cdot 3}$

Step 1.6.1.6.3

Factor $12$ out of $12 (- x) + 12 (30)$ .

$y = \frac{- 12 \pm \sqrt{12 (- x + 30)}}{2 \cdot 3}$

Step 1.6.1.7

Rewrite $12 (- x + 30)$ as $2^{2} \cdot (3 (- x + 30))$ .

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Step 1.6.1.7.1

Factor $4$ out of $12$ .

$y = \frac{- 12 \pm \sqrt{4 (3) (- x + 30)}}{2 \cdot 3}$

Step 1.6.1.7.2

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

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (3 (- x + 30))}}{2 \cdot 3}$

Step 1.6.1.7.3

Add parentheses.

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (3 (- x + 30))}}{2 \cdot 3}$

Step 1.6.1.8

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{3 (- x + 30)}}{2 \cdot 3}$

Step 1.6.2

Multiply $2$ by $3$ .

$y = \frac{- 12 \pm 2 \sqrt{3 (- x + 30)}}{6}$

Step 1.6.3

Simplify $\frac{- 12 \pm 2 \sqrt{3 (- x + 30)}}{6}$ .

$y = \frac{- 6 \pm \sqrt{3 (- x + 30)}}{3}$

Step 1.6.4

Change the $\pm$ to $-$ .

$y = \frac{- 6 - \sqrt{3 (- x + 30)}}{3}$

Step 1.6.5

Factor $- 1$ out of $- 6 - \sqrt{3 (- x + 30)}$ .

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Step 1.6.5.1

Rewrite $- 6$ as $- 1 (6)$ .

$y = \frac{- 1 \cdot 6 - \sqrt{3 (- x + 30)}}{3}$

Step 1.6.5.2

Factor $- 1$ out of $- \sqrt{3 (- x + 30)}$ .

$y = \frac{- 1 \cdot 6 - (\sqrt{3 (- x + 30)})}{3}$

Step 1.6.5.3

Factor $- 1$ out of $- 1 (6) - (\sqrt{3 (- x + 30)})$ .

$y = \frac{- 1 (6 + \sqrt{3 (- x + 30)})}{3}$

Step 1.6.5.4

Rewrite $- 1 (6 + \sqrt{3 (- x + 30)})$ as $- (6 + \sqrt{3 (- x + 30)})$ .

$y = \frac{- (6 + \sqrt{3 (- x + 30)})}{3}$

Step 1.6.6

Move the negative in front of the fraction.

$y = - \frac{6 + \sqrt{3 (- x + 30)}}{3}$

Step 1.7

The final answer is the combination of both solutions.

$y = - \frac{6 - \sqrt{3 (- x + 30)}}{3}$

$y = - \frac{6 + \sqrt{3 (- x + 30)}}{3}$

$y = - \frac{6 - \sqrt{3 (- x + 30)}}{3}$

$y = - \frac{6 + \sqrt{3 (- x + 30)}}{3}$

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

Split the fraction $\frac{6 - \sqrt{3 (- x + 30)}}{3}$ into two fractions.

$y = - (\frac{6}{3} + \frac{- \sqrt{3 (- x + 30)}}{3})$

$y = - \frac{6 + \sqrt{3 (- x + 30)}}{3}$

Step 4

Simplify each term.

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Step 4.1

Divide $6$ by $3$ .

$y = - (2 + \frac{- \sqrt{3 (- x + 30)}}{3})$

$y = - \frac{6 + \sqrt{3 (- x + 30)}}{3}$

Step 4.2

Move the negative in front of the fraction.

$y = - (2 - \frac{\sqrt{3 (- x + 30)}}{3})$

$y = - \frac{6 + \sqrt{3 (- x + 30)}}{3}$

$y = - (2 - \frac{\sqrt{3 (- x + 30)}}{3})$

$y = - \frac{6 + \sqrt{3 (- x + 30)}}{3}$

Step 5

Simplify by multiplying through.

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Step 5.1

Apply the distributive property.

$y = - 1 \cdot 2 + \frac{\sqrt{3 (- x + 30)}}{3}$

$y = - \frac{6 + \sqrt{3 (- x + 30)}}{3}$

Step 5.2

Multiply $- 1$ by $2$ .

$y = - 2 + \frac{\sqrt{3 (- x + 30)}}{3}$

$y = - \frac{6 + \sqrt{3 (- x + 30)}}{3}$

$y = - 2 + \frac{\sqrt{3 (- x + 30)}}{3}$

$y = - \frac{6 + \sqrt{3 (- x + 30)}}{3}$

Step 6

Split the fraction $\frac{6 + \sqrt{3 (- x + 30)}}{3}$ into two fractions.

$y = - 2 + \frac{\sqrt{3 (- x + 30)}}{3}$

$y = - (\frac{6}{3} + \frac{\sqrt{3 (- x + 30)}}{3})$

Step 7

Divide $6$ by $3$ .

$y = - 2 + \frac{\sqrt{3 (- x + 30)}}{3}$

$y = - (2 + \frac{\sqrt{3 (- x + 30)}}{3})$

Step 8

Apply the distributive property.

$y = - 2 + \frac{\sqrt{3 (- x + 30)}}{3}$

$y = - 1 \cdot 2 - \frac{\sqrt{3 (- x + 30)}}{3}$

Step 9

Multiply $- 1$ by $2$ .

$y = - 2 + \frac{\sqrt{3 (- x + 30)}}{3}$

$y = - 2 - \frac{\sqrt{3 (- x + 30)}}{3}$

Step 10

Reorder terms.

$y = - 2 + \frac{1}{3} \cdot \sqrt{3 (- x + 30)}$

$y = - 2 - (\frac{1}{3} \cdot \sqrt{3 (- x + 30)})$

Step 11

Remove parentheses.

$y = - 2 + \frac{1}{3} \cdot \sqrt{3 (- x + 30)}$

$y = - 2 - \frac{1}{3} \cdot \sqrt{3 (- x + 30)}$

Step 12