Pre-Algebra Examples

$3 x^{2} + y^{2} + 18 x - 2 y + 4 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = - 2$ , and $c = 3 x^{2} + 18 x + 4$ into the quadratic formula and solve for $y$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot (3 x^{2} + 18 x + 4))}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot (3 x^{2} + 18 x + 4)}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4$ by $1$ .

$y = \frac{2 \pm \sqrt{4 - 4 \cdot (3 x^{2} + 18 x + 4)}}{2 \cdot 1}$

Step 3.1.3

Apply the distributive property.

$y = \frac{2 \pm \sqrt{4 - 4 (3 x^{2}) - 4 (18 x) - 4 \cdot 4}}{2 \cdot 1}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Multiply $3$ by $- 4$ .

$y = \frac{2 \pm \sqrt{4 - 12 x^{2} - 4 (18 x) - 4 \cdot 4}}{2 \cdot 1}$

Step 3.1.4.2

Multiply $18$ by $- 4$ .

$y = \frac{2 \pm \sqrt{4 - 12 x^{2} - 72 x - 4 \cdot 4}}{2 \cdot 1}$

Step 3.1.4.3

Multiply $- 4$ by $4$ .

$y = \frac{2 \pm \sqrt{4 - 12 x^{2} - 72 x - 16}}{2 \cdot 1}$

Step 3.1.5

Subtract $16$ from $4$ .

$y = \frac{2 \pm \sqrt{- 12 x^{2} - 72 x - 12}}{2 \cdot 1}$

Step 3.1.6

Factor $12$ out of $- 12 x^{2} - 72 x - 12$ .

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Step 3.1.6.1

Factor $12$ out of $- 12 x^{2}$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2}) - 72 x - 12}}{2 \cdot 1}$

Step 3.1.6.2

Factor $12$ out of $- 72 x$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2}) + 12 (- 6 x) - 12}}{2 \cdot 1}$

Step 3.1.6.3

Factor $12$ out of $- 12$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2}) + 12 (- 6 x) + 12 (- 1)}}{2 \cdot 1}$

Step 3.1.6.4

Factor $12$ out of $12 (- x^{2}) + 12 (- 6 x)$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2} - 6 x) + 12 (- 1)}}{2 \cdot 1}$

Step 3.1.6.5

Factor $12$ out of $12 (- x^{2} - 6 x) + 12 (- 1)$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2} - 6 x - 1)}}{2 \cdot 1}$

Step 3.1.7

Rewrite $12 (- x^{2} - 6 x - 1)$ as $2^{2} \cdot (3 (- x^{2} - 6 x - 1))$ .

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Step 3.1.7.1

Factor $4$ out of $12$ .

$y = \frac{2 \pm \sqrt{4 (3) (- x^{2} - 6 x - 1)}}{2 \cdot 1}$

Step 3.1.7.2

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

$y = \frac{2 \pm \sqrt{2^{2} \cdot (3 (- x^{2} - 6 x - 1))}}{2 \cdot 1}$

Step 3.1.7.3

Add parentheses.

$y = \frac{2 \pm \sqrt{2^{2} \cdot (3 (- x^{2} - 6 x - 1))}}{2 \cdot 1}$

Step 3.1.8

Pull terms out from under the radical.

$y = \frac{2 \pm 2 \sqrt{3 (- x^{2} - 6 x - 1)}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$y = \frac{2 \pm 2 \sqrt{3 (- x^{2} - 6 x - 1)}}{2}$

Step 3.3

Simplify $\frac{2 \pm 2 \sqrt{3 (- x^{2} - 6 x - 1)}}{2}$ .

$y = 1 \pm \sqrt{3 (- x^{2} - 6 x - 1)}$

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot (3 x^{2} + 18 x + 4)}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4$ by $1$ .

$y = \frac{2 \pm \sqrt{4 - 4 \cdot (3 x^{2} + 18 x + 4)}}{2 \cdot 1}$

Step 4.1.3

Apply the distributive property.

$y = \frac{2 \pm \sqrt{4 - 4 (3 x^{2}) - 4 (18 x) - 4 \cdot 4}}{2 \cdot 1}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Multiply $3$ by $- 4$ .

$y = \frac{2 \pm \sqrt{4 - 12 x^{2} - 4 (18 x) - 4 \cdot 4}}{2 \cdot 1}$

Step 4.1.4.2

Multiply $18$ by $- 4$ .

$y = \frac{2 \pm \sqrt{4 - 12 x^{2} - 72 x - 4 \cdot 4}}{2 \cdot 1}$

Step 4.1.4.3

Multiply $- 4$ by $4$ .

$y = \frac{2 \pm \sqrt{4 - 12 x^{2} - 72 x - 16}}{2 \cdot 1}$

Step 4.1.5

Subtract $16$ from $4$ .

$y = \frac{2 \pm \sqrt{- 12 x^{2} - 72 x - 12}}{2 \cdot 1}$

Step 4.1.6

Factor $12$ out of $- 12 x^{2} - 72 x - 12$ .

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Step 4.1.6.1

Factor $12$ out of $- 12 x^{2}$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2}) - 72 x - 12}}{2 \cdot 1}$

Step 4.1.6.2

Factor $12$ out of $- 72 x$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2}) + 12 (- 6 x) - 12}}{2 \cdot 1}$

Step 4.1.6.3

Factor $12$ out of $- 12$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2}) + 12 (- 6 x) + 12 (- 1)}}{2 \cdot 1}$

Step 4.1.6.4

Factor $12$ out of $12 (- x^{2}) + 12 (- 6 x)$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2} - 6 x) + 12 (- 1)}}{2 \cdot 1}$

Step 4.1.6.5

Factor $12$ out of $12 (- x^{2} - 6 x) + 12 (- 1)$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2} - 6 x - 1)}}{2 \cdot 1}$

Step 4.1.7

Rewrite $12 (- x^{2} - 6 x - 1)$ as $2^{2} \cdot (3 (- x^{2} - 6 x - 1))$ .

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Step 4.1.7.1

Factor $4$ out of $12$ .

$y = \frac{2 \pm \sqrt{4 (3) (- x^{2} - 6 x - 1)}}{2 \cdot 1}$

Step 4.1.7.2

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

$y = \frac{2 \pm \sqrt{2^{2} \cdot (3 (- x^{2} - 6 x - 1))}}{2 \cdot 1}$

Step 4.1.7.3

Add parentheses.

$y = \frac{2 \pm \sqrt{2^{2} \cdot (3 (- x^{2} - 6 x - 1))}}{2 \cdot 1}$

Step 4.1.8

Pull terms out from under the radical.

$y = \frac{2 \pm 2 \sqrt{3 (- x^{2} - 6 x - 1)}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$y = \frac{2 \pm 2 \sqrt{3 (- x^{2} - 6 x - 1)}}{2}$

Step 4.3

Simplify $\frac{2 \pm 2 \sqrt{3 (- x^{2} - 6 x - 1)}}{2}$ .

$y = 1 \pm \sqrt{3 (- x^{2} - 6 x - 1)}$

Step 4.4

Change the $\pm$ to $+$ .

$y = 1 + \sqrt{3 (- x^{2} - 6 x - 1)}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot (3 x^{2} + 18 x + 4)}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4$ by $1$ .

$y = \frac{2 \pm \sqrt{4 - 4 \cdot (3 x^{2} + 18 x + 4)}}{2 \cdot 1}$

Step 5.1.3

Apply the distributive property.

$y = \frac{2 \pm \sqrt{4 - 4 (3 x^{2}) - 4 (18 x) - 4 \cdot 4}}{2 \cdot 1}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Multiply $3$ by $- 4$ .

$y = \frac{2 \pm \sqrt{4 - 12 x^{2} - 4 (18 x) - 4 \cdot 4}}{2 \cdot 1}$

Step 5.1.4.2

Multiply $18$ by $- 4$ .

$y = \frac{2 \pm \sqrt{4 - 12 x^{2} - 72 x - 4 \cdot 4}}{2 \cdot 1}$

Step 5.1.4.3

Multiply $- 4$ by $4$ .

$y = \frac{2 \pm \sqrt{4 - 12 x^{2} - 72 x - 16}}{2 \cdot 1}$

Step 5.1.5

Subtract $16$ from $4$ .

$y = \frac{2 \pm \sqrt{- 12 x^{2} - 72 x - 12}}{2 \cdot 1}$

Step 5.1.6

Factor $12$ out of $- 12 x^{2} - 72 x - 12$ .

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Step 5.1.6.1

Factor $12$ out of $- 12 x^{2}$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2}) - 72 x - 12}}{2 \cdot 1}$

Step 5.1.6.2

Factor $12$ out of $- 72 x$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2}) + 12 (- 6 x) - 12}}{2 \cdot 1}$

Step 5.1.6.3

Factor $12$ out of $- 12$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2}) + 12 (- 6 x) + 12 (- 1)}}{2 \cdot 1}$

Step 5.1.6.4

Factor $12$ out of $12 (- x^{2}) + 12 (- 6 x)$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2} - 6 x) + 12 (- 1)}}{2 \cdot 1}$

Step 5.1.6.5

Factor $12$ out of $12 (- x^{2} - 6 x) + 12 (- 1)$ .

$y = \frac{2 \pm \sqrt{12 (- x^{2} - 6 x - 1)}}{2 \cdot 1}$

Step 5.1.7

Rewrite $12 (- x^{2} - 6 x - 1)$ as $2^{2} \cdot (3 (- x^{2} - 6 x - 1))$ .

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Step 5.1.7.1

Factor $4$ out of $12$ .

$y = \frac{2 \pm \sqrt{4 (3) (- x^{2} - 6 x - 1)}}{2 \cdot 1}$

Step 5.1.7.2

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

$y = \frac{2 \pm \sqrt{2^{2} \cdot (3 (- x^{2} - 6 x - 1))}}{2 \cdot 1}$

Step 5.1.7.3

Add parentheses.

$y = \frac{2 \pm \sqrt{2^{2} \cdot (3 (- x^{2} - 6 x - 1))}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical.

$y = \frac{2 \pm 2 \sqrt{3 (- x^{2} - 6 x - 1)}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$y = \frac{2 \pm 2 \sqrt{3 (- x^{2} - 6 x - 1)}}{2}$

Step 5.3

Simplify $\frac{2 \pm 2 \sqrt{3 (- x^{2} - 6 x - 1)}}{2}$ .

$y = 1 \pm \sqrt{3 (- x^{2} - 6 x - 1)}$

Step 5.4

Change the $\pm$ to $-$ .

$y = 1 - \sqrt{3 (- x^{2} - 6 x - 1)}$

Step 6

The final answer is the combination of both solutions.

$y = 1 + \sqrt{3 (- x^{2} - 6 x - 1)}$

$y = 1 - \sqrt{3 (- x^{2} - 6 x - 1)}$

Step 7

The given equation $y = 1 + \sqrt{3 (- x^{2} - 6 x - 1)}, 1 - \sqrt{3 (- x^{2} - 6 x - 1)}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x$