Pre-Algebra Examples

$36 x^{2} + 81 y^{2} + 504 x - 324 y - 828 = 0$

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 $a = 81$ , $b = - 324$ , and $c = 36 x^{2} + 504 x - 828$ into the quadratic formula and solve for $y$ .

$\frac{324 \pm \sqrt{{(- 324)}^{2} - 4 \cdot (81 \cdot (36 x^{2} + 504 x - 828))}}{2 \cdot 81}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 324$ to the power of $2$ .

$y = \frac{324 \pm \sqrt{104976 - 4 \cdot 81 \cdot (36 x^{2} + 504 x - 828)}}{2 \cdot 81}$

Step 3.1.2

Multiply $- 4$ by $81$ .

$y = \frac{324 \pm \sqrt{104976 - 324 \cdot (36 x^{2} + 504 x - 828)}}{2 \cdot 81}$

Step 3.1.3

Apply the distributive property.

$y = \frac{324 \pm \sqrt{104976 - 324 (36 x^{2}) - 324 (504 x) - 324 \cdot - 828}}{2 \cdot 81}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Multiply $36$ by $- 324$ .

$y = \frac{324 \pm \sqrt{104976 - 11664 x^{2} - 324 (504 x) - 324 \cdot - 828}}{2 \cdot 81}$

Step 3.1.4.2

Multiply $504$ by $- 324$ .

$y = \frac{324 \pm \sqrt{104976 - 11664 x^{2} - 163296 x - 324 \cdot - 828}}{2 \cdot 81}$

Step 3.1.4.3

Multiply $- 324$ by $- 828$ .

$y = \frac{324 \pm \sqrt{104976 - 11664 x^{2} - 163296 x + 268272}}{2 \cdot 81}$

Step 3.1.5

Add $104976$ and $268272$ .

$y = \frac{324 \pm \sqrt{- 11664 x^{2} - 163296 x + 373248}}{2 \cdot 81}$

Step 3.1.6

Rewrite $- 11664 x^{2} - 163296 x + 373248$ in a factored form.

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Step 3.1.6.1

Factor $11664$ out of $- 11664 x^{2} - 163296 x + 373248$ .

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Step 3.1.6.1.1

Factor $11664$ out of $- 11664 x^{2}$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2}) - 163296 x + 373248}}{2 \cdot 81}$

Step 3.1.6.1.2

Factor $11664$ out of $- 163296 x$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2}) + 11664 (- 14 x) + 373248}}{2 \cdot 81}$

Step 3.1.6.1.3

Factor $11664$ out of $373248$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2}) + 11664 (- 14 x) + 11664 (32)}}{2 \cdot 81}$

Step 3.1.6.1.4

Factor $11664$ out of $11664 (- x^{2}) + 11664 (- 14 x)$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2} - 14 x) + 11664 (32)}}{2 \cdot 81}$

Step 3.1.6.1.5

Factor $11664$ out of $11664 (- x^{2} - 14 x) + 11664 (32)$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2} - 14 x + 32)}}{2 \cdot 81}$

Step 3.1.6.2

Factor by grouping.

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Step 3.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 32 = - 32$ and whose sum is $b = - 14$ .

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Step 3.1.6.2.1.1

Factor $- 14$ out of $- 14 x$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2} - 14 x + 32)}}{2 \cdot 81}$

Step 3.1.6.2.1.2

Rewrite $- 14$ as $2$ plus $- 16$

$y = \frac{324 \pm \sqrt{11664 (- x^{2} + (2 - 16) x + 32)}}{2 \cdot 81}$

Step 3.1.6.2.1.3

Apply the distributive property.

$y = \frac{324 \pm \sqrt{11664 (- x^{2} + 2 x - 16 x + 32)}}{2 \cdot 81}$

Step 3.1.6.2.2

Factor out the greatest common factor from each group.

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Step 3.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{324 \pm \sqrt{11664 ((- x^{2} + 2 x) - 16 x + 32)}}{2 \cdot 81}$

Step 3.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{324 \pm \sqrt{11664 (x (- x + 2) + 16 (- x + 2))}}{2 \cdot 81}$

Step 3.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

$y = \frac{324 \pm \sqrt{11664 ((- x + 2) (x + 16))}}{2 \cdot 81}$

$y = \frac{324 \pm \sqrt{11664 (- x + 2) (x + 16)}}{2 \cdot 81}$

Step 3.1.7

Rewrite $11664 (- x + 2) (x + 16)$ as $108^{2} ((- x + 2) (x + 4^{2}))$ .

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Step 3.1.7.1

Rewrite $11664$ as $108^{2}$ .

$y = \frac{324 \pm \sqrt{108^{2} (- x + 2) (x + 16)}}{2 \cdot 81}$

Step 3.1.7.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{324 \pm \sqrt{108^{2} (- x + 2) (x + 4^{2})}}{2 \cdot 81}$

Step 3.1.7.3

Add parentheses.

$y = \frac{324 \pm \sqrt{108^{2} ((- x + 2) (x + 4^{2}))}}{2 \cdot 81}$

Step 3.1.8

Pull terms out from under the radical.

$y = \frac{324 \pm 108 \sqrt{(- x + 2) (x + 4^{2})}}{2 \cdot 81}$

Step 3.1.9

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

$y = \frac{324 \pm 108 \sqrt{(- x + 2) (x + 16)}}{2 \cdot 81}$

Step 3.2

Multiply $2$ by $81$ .

$y = \frac{324 \pm 108 \sqrt{(- x + 2) (x + 16)}}{162}$

Step 3.3

Simplify $\frac{324 \pm 108 \sqrt{(- x + 2) (x + 16)}}{162}$ .

$y = \frac{6 \pm 2 \sqrt{(- x + 2) (x + 16)}}{3}$

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 $- 324$ to the power of $2$ .

$y = \frac{324 \pm \sqrt{104976 - 4 \cdot 81 \cdot (36 x^{2} + 504 x - 828)}}{2 \cdot 81}$

Step 4.1.2

Multiply $- 4$ by $81$ .

$y = \frac{324 \pm \sqrt{104976 - 324 \cdot (36 x^{2} + 504 x - 828)}}{2 \cdot 81}$

Step 4.1.3

Apply the distributive property.

$y = \frac{324 \pm \sqrt{104976 - 324 (36 x^{2}) - 324 (504 x) - 324 \cdot - 828}}{2 \cdot 81}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Multiply $36$ by $- 324$ .

$y = \frac{324 \pm \sqrt{104976 - 11664 x^{2} - 324 (504 x) - 324 \cdot - 828}}{2 \cdot 81}$

Step 4.1.4.2

Multiply $504$ by $- 324$ .

$y = \frac{324 \pm \sqrt{104976 - 11664 x^{2} - 163296 x - 324 \cdot - 828}}{2 \cdot 81}$

Step 4.1.4.3

Multiply $- 324$ by $- 828$ .

$y = \frac{324 \pm \sqrt{104976 - 11664 x^{2} - 163296 x + 268272}}{2 \cdot 81}$

Step 4.1.5

Add $104976$ and $268272$ .

$y = \frac{324 \pm \sqrt{- 11664 x^{2} - 163296 x + 373248}}{2 \cdot 81}$

Step 4.1.6

Rewrite $- 11664 x^{2} - 163296 x + 373248$ in a factored form.

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Step 4.1.6.1

Factor $11664$ out of $- 11664 x^{2} - 163296 x + 373248$ .

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Step 4.1.6.1.1

Factor $11664$ out of $- 11664 x^{2}$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2}) - 163296 x + 373248}}{2 \cdot 81}$

Step 4.1.6.1.2

Factor $11664$ out of $- 163296 x$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2}) + 11664 (- 14 x) + 373248}}{2 \cdot 81}$

Step 4.1.6.1.3

Factor $11664$ out of $373248$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2}) + 11664 (- 14 x) + 11664 (32)}}{2 \cdot 81}$

Step 4.1.6.1.4

Factor $11664$ out of $11664 (- x^{2}) + 11664 (- 14 x)$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2} - 14 x) + 11664 (32)}}{2 \cdot 81}$

Step 4.1.6.1.5

Factor $11664$ out of $11664 (- x^{2} - 14 x) + 11664 (32)$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2} - 14 x + 32)}}{2 \cdot 81}$

Step 4.1.6.2

Factor by grouping.

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Step 4.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 32 = - 32$ and whose sum is $b = - 14$ .

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Step 4.1.6.2.1.1

Factor $- 14$ out of $- 14 x$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2} - 14 x + 32)}}{2 \cdot 81}$

Step 4.1.6.2.1.2

Rewrite $- 14$ as $2$ plus $- 16$

$y = \frac{324 \pm \sqrt{11664 (- x^{2} + (2 - 16) x + 32)}}{2 \cdot 81}$

Step 4.1.6.2.1.3

Apply the distributive property.

$y = \frac{324 \pm \sqrt{11664 (- x^{2} + 2 x - 16 x + 32)}}{2 \cdot 81}$

Step 4.1.6.2.2

Factor out the greatest common factor from each group.

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Step 4.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{324 \pm \sqrt{11664 ((- x^{2} + 2 x) - 16 x + 32)}}{2 \cdot 81}$

Step 4.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{324 \pm \sqrt{11664 (x (- x + 2) + 16 (- x + 2))}}{2 \cdot 81}$

Step 4.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

$y = \frac{324 \pm \sqrt{11664 ((- x + 2) (x + 16))}}{2 \cdot 81}$

$y = \frac{324 \pm \sqrt{11664 (- x + 2) (x + 16)}}{2 \cdot 81}$

Step 4.1.7

Rewrite $11664 (- x + 2) (x + 16)$ as $108^{2} ((- x + 2) (x + 4^{2}))$ .

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Step 4.1.7.1

Rewrite $11664$ as $108^{2}$ .

$y = \frac{324 \pm \sqrt{108^{2} (- x + 2) (x + 16)}}{2 \cdot 81}$

Step 4.1.7.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{324 \pm \sqrt{108^{2} (- x + 2) (x + 4^{2})}}{2 \cdot 81}$

Step 4.1.7.3

Add parentheses.

$y = \frac{324 \pm \sqrt{108^{2} ((- x + 2) (x + 4^{2}))}}{2 \cdot 81}$

Step 4.1.8

Pull terms out from under the radical.

$y = \frac{324 \pm 108 \sqrt{(- x + 2) (x + 4^{2})}}{2 \cdot 81}$

Step 4.1.9

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

$y = \frac{324 \pm 108 \sqrt{(- x + 2) (x + 16)}}{2 \cdot 81}$

Step 4.2

Multiply $2$ by $81$ .

$y = \frac{324 \pm 108 \sqrt{(- x + 2) (x + 16)}}{162}$

Step 4.3

Simplify $\frac{324 \pm 108 \sqrt{(- x + 2) (x + 16)}}{162}$ .

$y = \frac{6 \pm 2 \sqrt{(- x + 2) (x + 16)}}{3}$

Step 4.4

Change the $\pm$ to $+$ .

$y = \frac{6 + 2 \sqrt{(- x + 2) (x + 16)}}{3}$

Step 4.5

Factor $2$ out of $6 + 2 \sqrt{(- x + 2) (x + 16)}$ .

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Step 4.5.1

Factor $2$ out of $6$ .

$y = \frac{2 \cdot 3 + 2 \sqrt{(- x + 2) (x + 16)}}{3}$

Step 4.5.2

Factor $2$ out of $2 \cdot 3 + 2 \sqrt{(- x + 2) (x + 16)}$ .

$y = \frac{2 (3 + \sqrt{(- x + 2) (x + 16)})}{3}$

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 $- 324$ to the power of $2$ .

$y = \frac{324 \pm \sqrt{104976 - 4 \cdot 81 \cdot (36 x^{2} + 504 x - 828)}}{2 \cdot 81}$

Step 5.1.2

Multiply $- 4$ by $81$ .

$y = \frac{324 \pm \sqrt{104976 - 324 \cdot (36 x^{2} + 504 x - 828)}}{2 \cdot 81}$

Step 5.1.3

Apply the distributive property.

$y = \frac{324 \pm \sqrt{104976 - 324 (36 x^{2}) - 324 (504 x) - 324 \cdot - 828}}{2 \cdot 81}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Multiply $36$ by $- 324$ .

$y = \frac{324 \pm \sqrt{104976 - 11664 x^{2} - 324 (504 x) - 324 \cdot - 828}}{2 \cdot 81}$

Step 5.1.4.2

Multiply $504$ by $- 324$ .

$y = \frac{324 \pm \sqrt{104976 - 11664 x^{2} - 163296 x - 324 \cdot - 828}}{2 \cdot 81}$

Step 5.1.4.3

Multiply $- 324$ by $- 828$ .

$y = \frac{324 \pm \sqrt{104976 - 11664 x^{2} - 163296 x + 268272}}{2 \cdot 81}$

Step 5.1.5

Add $104976$ and $268272$ .

$y = \frac{324 \pm \sqrt{- 11664 x^{2} - 163296 x + 373248}}{2 \cdot 81}$

Step 5.1.6

Rewrite $- 11664 x^{2} - 163296 x + 373248$ in a factored form.

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Step 5.1.6.1

Factor $11664$ out of $- 11664 x^{2} - 163296 x + 373248$ .

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Step 5.1.6.1.1

Factor $11664$ out of $- 11664 x^{2}$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2}) - 163296 x + 373248}}{2 \cdot 81}$

Step 5.1.6.1.2

Factor $11664$ out of $- 163296 x$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2}) + 11664 (- 14 x) + 373248}}{2 \cdot 81}$

Step 5.1.6.1.3

Factor $11664$ out of $373248$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2}) + 11664 (- 14 x) + 11664 (32)}}{2 \cdot 81}$

Step 5.1.6.1.4

Factor $11664$ out of $11664 (- x^{2}) + 11664 (- 14 x)$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2} - 14 x) + 11664 (32)}}{2 \cdot 81}$

Step 5.1.6.1.5

Factor $11664$ out of $11664 (- x^{2} - 14 x) + 11664 (32)$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2} - 14 x + 32)}}{2 \cdot 81}$

Step 5.1.6.2

Factor by grouping.

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Step 5.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 32 = - 32$ and whose sum is $b = - 14$ .

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Step 5.1.6.2.1.1

Factor $- 14$ out of $- 14 x$ .

$y = \frac{324 \pm \sqrt{11664 (- x^{2} - 14 x + 32)}}{2 \cdot 81}$

Step 5.1.6.2.1.2

Rewrite $- 14$ as $2$ plus $- 16$

$y = \frac{324 \pm \sqrt{11664 (- x^{2} + (2 - 16) x + 32)}}{2 \cdot 81}$

Step 5.1.6.2.1.3

Apply the distributive property.

$y = \frac{324 \pm \sqrt{11664 (- x^{2} + 2 x - 16 x + 32)}}{2 \cdot 81}$

Step 5.1.6.2.2

Factor out the greatest common factor from each group.

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Step 5.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{324 \pm \sqrt{11664 ((- x^{2} + 2 x) - 16 x + 32)}}{2 \cdot 81}$

Step 5.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{324 \pm \sqrt{11664 (x (- x + 2) + 16 (- x + 2))}}{2 \cdot 81}$

Step 5.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

$y = \frac{324 \pm \sqrt{11664 ((- x + 2) (x + 16))}}{2 \cdot 81}$

$y = \frac{324 \pm \sqrt{11664 (- x + 2) (x + 16)}}{2 \cdot 81}$

Step 5.1.7

Rewrite $11664 (- x + 2) (x + 16)$ as $108^{2} ((- x + 2) (x + 4^{2}))$ .

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Step 5.1.7.1

Rewrite $11664$ as $108^{2}$ .

$y = \frac{324 \pm \sqrt{108^{2} (- x + 2) (x + 16)}}{2 \cdot 81}$

Step 5.1.7.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{324 \pm \sqrt{108^{2} (- x + 2) (x + 4^{2})}}{2 \cdot 81}$

Step 5.1.7.3

Add parentheses.

$y = \frac{324 \pm \sqrt{108^{2} ((- x + 2) (x + 4^{2}))}}{2 \cdot 81}$

Step 5.1.8

Pull terms out from under the radical.

$y = \frac{324 \pm 108 \sqrt{(- x + 2) (x + 4^{2})}}{2 \cdot 81}$

Step 5.1.9

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

$y = \frac{324 \pm 108 \sqrt{(- x + 2) (x + 16)}}{2 \cdot 81}$

Step 5.2

Multiply $2$ by $81$ .

$y = \frac{324 \pm 108 \sqrt{(- x + 2) (x + 16)}}{162}$

Step 5.3

Simplify $\frac{324 \pm 108 \sqrt{(- x + 2) (x + 16)}}{162}$ .

$y = \frac{6 \pm 2 \sqrt{(- x + 2) (x + 16)}}{3}$

Step 5.4

Change the $\pm$ to $-$ .

$y = \frac{6 - 2 \sqrt{(- x + 2) (x + 16)}}{3}$

Step 5.5

Factor $2$ out of $6 - 2 \sqrt{(- x + 2) (x + 16)}$ .

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Step 5.5.1

Factor $2$ out of $6$ .

$y = \frac{2 (3) - 2 \sqrt{(- x + 2) (x + 16)}}{3}$

Step 5.5.2

Factor $2$ out of $- 2 \sqrt{(- x + 2) (x + 16)}$ .

$y = \frac{2 (3) + 2 (- \sqrt{(- x + 2) (x + 16)})}{3}$

Step 5.5.3

Factor $2$ out of $2 (3) + 2 (- \sqrt{(- x + 2) (x + 16)})$ .

$y = \frac{2 (3 - \sqrt{(- x + 2) (x + 16)})}{3}$

Step 6

The final answer is the combination of both solutions.

$y = \frac{2 (3 + \sqrt{(- x + 2) (x + 16)})}{3}$

$y = \frac{2 (3 - \sqrt{(- x + 2) (x + 16)})}{3}$

Step 7

The given equation $y = \frac{2 (3 + \sqrt{(- x + 2) (x + 16)})}{3}, \frac{2 (3 - \sqrt{(- x + 2) (x + 16)})}{3}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

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