Algebra Examples

$\frac{1}{3} y - x^{2} + 2 x = - 5$

Step 1

Combine $\frac{1}{3}$ and $y$ .

$\frac{y}{3} - x^{2} + 2 x = - 5$

Step 2

Add $5$ to both sides of the equation.

$\frac{y}{3} - x^{2} + 2 x + 5 = 0$

Step 3

Multiply through by the least common denominator $3$ , then simplify.

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Step 3.1

Apply the distributive property.

$3 (\frac{y}{3}) + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 5 = 0$

Step 3.2

Simplify.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

$3 (\frac{y}{3}) + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 5 = 0$

Step 3.2.1.2

Rewrite the expression.

$y + 3 (- x^{2}) + 3 (2 x) + 3 \cdot 5 = 0$

Step 3.2.2

Multiply $- 1$ by $3$ .

$y - 3 x^{2} + 3 (2 x) + 3 \cdot 5 = 0$

Step 3.2.3

Multiply $2$ by $3$ .

$y - 3 x^{2} + 6 x + 3 \cdot 5 = 0$

Step 3.2.4

Multiply $3$ by $5$ .

$y - 3 x^{2} + 6 x + 15 = 0$

Step 3.3

Reorder $y$ and $- 3 x^{2}$ .

$- 3 x^{2} + y + 6 x + 15 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = - 3$ , $b = 6$ , and $c = y + 15$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 3 \cdot (y + 15))}}{2 \cdot - 3}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 3 \cdot (y + 15)}}{2 \cdot - 3}$

Step 6.1.2

Multiply $- 4$ by $- 3$ .

$x = \frac{- 6 \pm \sqrt{36 + 12 \cdot (y + 15)}}{2 \cdot - 3}$

Step 6.1.3

Apply the distributive property.

$x = \frac{- 6 \pm \sqrt{36 + 12 y + 12 \cdot 15}}{2 \cdot - 3}$

Step 6.1.4

Multiply $12$ by $15$ .

$x = \frac{- 6 \pm \sqrt{36 + 12 y + 180}}{2 \cdot - 3}$

Step 6.1.5

Add $36$ and $180$ .

$x = \frac{- 6 \pm \sqrt{12 y + 216}}{2 \cdot - 3}$

Step 6.1.6

Factor $12$ out of $12 y + 216$ .

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Step 6.1.6.1

Factor $12$ out of $12 y$ .

$x = \frac{- 6 \pm \sqrt{12 (y) + 216}}{2 \cdot - 3}$

Step 6.1.6.2

Factor $12$ out of $216$ .

$x = \frac{- 6 \pm \sqrt{12 y + 12 \cdot 18}}{2 \cdot - 3}$

Step 6.1.6.3

Factor $12$ out of $12 y + 12 \cdot 18$ .

$x = \frac{- 6 \pm \sqrt{12 (y + 18)}}{2 \cdot - 3}$

Step 6.1.7

Rewrite $12 (y + 18)$ as $2^{2} \cdot (3 (y + 18))$ .

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Step 6.1.7.1

Factor $4$ out of $12$ .

$x = \frac{- 6 \pm \sqrt{4 (3) (y + 18)}}{2 \cdot - 3}$

Step 6.1.7.2

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot (3 (y + 18))}}{2 \cdot - 3}$

Step 6.1.7.3

Add parentheses.

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot (3 (y + 18))}}{2 \cdot - 3}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{3 (y + 18)}}{2 \cdot - 3}$

Step 6.2

Multiply $2$ by $- 3$ .

$x = \frac{- 6 \pm 2 \sqrt{3 (y + 18)}}{- 6}$

Step 6.3

Simplify $\frac{- 6 \pm 2 \sqrt{3 (y + 18)}}{- 6}$ .

$x = \frac{3 \pm \sqrt{3 (y + 18)}}{3}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{3 (y + 18)}}{3}$

$x = \frac{3 - \sqrt{3 (y + 18)}}{3}$