Precalculus Examples

$25 x^{2} - 120 x y + 144 y^{2} - 624 x - 260 y + 676 = 0$

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 $a = 25$ , $b = - 120 y - 624$ , and $c = 144 y^{2} - 260 y + 676$ into the quadratic formula and solve for $x$ .

$\frac{- (- 120 y - 624) \pm \sqrt{{(- 120 y - 624)}^{2} - 4 \cdot (25 \cdot (144 y^{2} - 260 y + 676))}}{2 \cdot 25}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Apply the distributive property.

$x = \frac{- (- 120 y) + 624 \pm \sqrt{{(- 120 y - 624)}^{2} - 4 \cdot 25 \cdot (144 y^{2} - 260 y + 676)}}{2 \cdot 25}$

Step 3.1.2

Multiply $- 120$ by $- 1$ .

$x = \frac{120 y + 624 \pm \sqrt{{(- 120 y - 624)}^{2} - 4 \cdot 25 \cdot (144 y^{2} - 260 y + 676)}}{2 \cdot 25}$

Step 3.1.3

Multiply $- 1$ by $- 624$ .

$x = \frac{120 y + 624 \pm \sqrt{{(- 120 y - 624)}^{2} - 4 \cdot 25 \cdot (144 y^{2} - 260 y + 676)}}{2 \cdot 25}$

Step 3.1.4

Add parentheses.

$x = \frac{120 y + 624 \pm \sqrt{{(- 120 y - 624)}^{2} - 4 \cdot (25 \cdot (144 y^{2} - 260 y + 676))}}{2 \cdot 25}$

Step 3.1.5

Let $u = 25 \cdot (144 y^{2} - 260 y + 676)$ . Substitute $u$ for all occurrences of $25 \cdot (144 y^{2} - 260 y + 676)$ .

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Step 3.1.5.1

Rewrite ${(- 120 y - 624)}^{2}$ as $(- 120 y - 624) (- 120 y - 624)$ .

$x = \frac{120 y + 624 \pm \sqrt{(- 120 y - 624) (- 120 y - 624) - 4 \cdot u}}{2 \cdot 25}$

Step 3.1.5.2

Expand $(- 120 y - 624) (- 120 y - 624)$ using the FOIL Method.

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Step 3.1.5.2.1

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{- 120 y (- 120 y - 624) - 624 (- 120 y - 624) - 4 \cdot u}}{2 \cdot 25}$

Step 3.1.5.2.2

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{- 120 y (- 120 y) - 120 y \cdot - 624 - 624 (- 120 y - 624) - 4 \cdot u}}{2 \cdot 25}$

Step 3.1.5.2.3

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{- 120 y (- 120 y) - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 3.1.5.3

Simplify and combine like terms.

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Step 3.1.5.3.1

Simplify each term.

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Step 3.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{120 y + 624 \pm \sqrt{- 120 \cdot (- 120 y \cdot y) - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 3.1.5.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 3.1.5.3.1.2.1

Move $y$ .

$x = \frac{120 y + 624 \pm \sqrt{- 120 \cdot (- 120 (y \cdot y)) - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 3.1.5.3.1.2.2

Multiply $y$ by $y$ .

$x = \frac{120 y + 624 \pm \sqrt{- 120 \cdot (- 120 y^{2}) - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 3.1.5.3.1.3

Multiply $- 120$ by $- 120$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 3.1.5.3.1.4

Multiply $- 624$ by $- 120$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 74880 y - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 3.1.5.3.1.5

Multiply $- 120$ by $- 624$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 74880 y + 74880 y - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 3.1.5.3.1.6

Multiply $- 624$ by $- 624$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 74880 y + 74880 y + 389376 - 4 \cdot u}}{2 \cdot 25}$

Step 3.1.5.3.2

Add $74880 y$ and $74880 y$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 149760 y + 389376 - 4 \cdot u}}{2 \cdot 25}$

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 149760 y + 389376 - 4 u}}{2 \cdot 25}$

Step 3.1.6

Factor $4$ out of $14400 y^{2} + 149760 y + 389376 - 4 u$ .

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Step 3.1.6.1

Factor $4$ out of $14400 y^{2}$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2}) + 149760 y + 389376 - 4 u}}{2 \cdot 25}$

Step 3.1.6.2

Factor $4$ out of $149760 y$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2}) + 4 (37440 y) + 389376 - 4 u}}{2 \cdot 25}$

Step 3.1.6.3

Factor $4$ out of $389376$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2}) + 4 (37440 y) + 4 (97344) - 4 u}}{2 \cdot 25}$

Step 3.1.6.4

Factor $4$ out of $- 4 u$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2}) + 4 (37440 y) + 4 (97344) + 4 (- u)}}{2 \cdot 25}$

Step 3.1.6.5

Factor $4$ out of $4 (3600 y^{2}) + 4 (37440 y)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y) + 4 (97344) + 4 (- u)}}{2 \cdot 25}$

Step 3.1.6.6

Factor $4$ out of $4 (3600 y^{2} + 37440 y) + 4 (97344)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344) + 4 (- u)}}{2 \cdot 25}$

Step 3.1.6.7

Factor $4$ out of $4 (3600 y^{2} + 37440 y + 97344) + 4 (- u)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - u)}}{2 \cdot 25}$

Step 3.1.7

Replace all occurrences of $u$ with $25 \cdot (144 y^{2} - 260 y + 676)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (25 \cdot (144 y^{2} - 260 y + 676)))}}{2 \cdot 25}$

Step 3.1.8

Simplify.

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Step 3.1.8.1

Simplify each term.

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Step 3.1.8.1.1

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (25 (144 y^{2}) + 25 (- 260 y) + 25 \cdot 676))}}{2 \cdot 25}$

Step 3.1.8.1.2

Simplify.

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Step 3.1.8.1.2.1

Multiply $144$ by $25$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (3600 y^{2} + 25 (- 260 y) + 25 \cdot 676))}}{2 \cdot 25}$

Step 3.1.8.1.2.2

Multiply $- 260$ by $25$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (3600 y^{2} - 6500 y + 25 \cdot 676))}}{2 \cdot 25}$

Step 3.1.8.1.2.3

Multiply $25$ by $676$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (3600 y^{2} - 6500 y + 16900))}}{2 \cdot 25}$

Step 3.1.8.1.3

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (3600 y^{2}) - (- 6500 y) - 1 \cdot 16900)}}{2 \cdot 25}$

Step 3.1.8.1.4

Simplify.

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Step 3.1.8.1.4.1

Multiply $3600$ by $- 1$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - 3600 y^{2} - (- 6500 y) - 1 \cdot 16900)}}{2 \cdot 25}$

Step 3.1.8.1.4.2

Multiply $- 6500$ by $- 1$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - 3600 y^{2} + 6500 y - 1 \cdot 16900)}}{2 \cdot 25}$

Step 3.1.8.1.4.3

Multiply $- 1$ by $16900$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - 3600 y^{2} + 6500 y - 16900)}}{2 \cdot 25}$

Step 3.1.8.2

Combine the opposite terms in $3600 y^{2} + 37440 y + 97344 - 3600 y^{2} + 6500 y - 16900$ .

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Step 3.1.8.2.1

Subtract $3600 y^{2}$ from $3600 y^{2}$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (37440 y + 97344 + 0 + 6500 y - 16900)}}{2 \cdot 25}$

Step 3.1.8.2.2

Add $37440 y + 97344$ and $0$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (37440 y + 97344 + 6500 y - 16900)}}{2 \cdot 25}$

Step 3.1.8.3

Add $37440 y$ and $6500 y$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (43940 y + 97344 - 16900)}}{2 \cdot 25}$

Step 3.1.8.4

Subtract $16900$ from $97344$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (43940 y + 80444)}}{2 \cdot 25}$

Step 3.1.9

Factor $676$ out of $43940 y + 80444$ .

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Step 3.1.9.1

Factor $676$ out of $43940 y$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (676 (65 y) + 80444)}}{2 \cdot 25}$

Step 3.1.9.2

Factor $676$ out of $80444$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (676 (65 y) + 676 (119))}}{2 \cdot 25}$

Step 3.1.9.3

Factor $676$ out of $676 (65 y) + 676 (119)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (676 (65 y + 119))}}{2 \cdot 25}$

$x = \frac{120 y + 624 \pm \sqrt{4 \cdot (676 (65 y + 119))}}{2 \cdot 25}$

Step 3.1.10

Multiply $4$ by $676$ .

$x = \frac{120 y + 624 \pm \sqrt{2704 (65 y + 119)}}{2 \cdot 25}$

Step 3.1.11

Rewrite $2704$ as $52^{2}$ .

$x = \frac{120 y + 624 \pm \sqrt{52^{2} (65 y + 119)}}{2 \cdot 25}$

Step 3.1.12

Pull terms out from under the radical.

$x = \frac{120 y + 624 \pm 52 \sqrt{65 y + 119}}{2 \cdot 25}$

Step 3.2

Multiply $2$ by $25$ .

$x = \frac{120 y + 624 \pm 52 \sqrt{65 y + 119}}{50}$

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

Apply the distributive property.

$x = \frac{- (- 120 y) + 624 \pm \sqrt{{(- 120 y - 624)}^{2} - 4 \cdot 25 \cdot (144 y^{2} - 260 y + 676)}}{2 \cdot 25}$

Step 4.1.2

Multiply $- 120$ by $- 1$ .

$x = \frac{120 y + 624 \pm \sqrt{{(- 120 y - 624)}^{2} - 4 \cdot 25 \cdot (144 y^{2} - 260 y + 676)}}{2 \cdot 25}$

Step 4.1.3

Multiply $- 1$ by $- 624$ .

$x = \frac{120 y + 624 \pm \sqrt{{(- 120 y - 624)}^{2} - 4 \cdot 25 \cdot (144 y^{2} - 260 y + 676)}}{2 \cdot 25}$

Step 4.1.4

Add parentheses.

$x = \frac{120 y + 624 \pm \sqrt{{(- 120 y - 624)}^{2} - 4 \cdot (25 \cdot (144 y^{2} - 260 y + 676))}}{2 \cdot 25}$

Step 4.1.5

Let $u = 25 \cdot (144 y^{2} - 260 y + 676)$ . Substitute $u$ for all occurrences of $25 \cdot (144 y^{2} - 260 y + 676)$ .

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Step 4.1.5.1

Rewrite ${(- 120 y - 624)}^{2}$ as $(- 120 y - 624) (- 120 y - 624)$ .

$x = \frac{120 y + 624 \pm \sqrt{(- 120 y - 624) (- 120 y - 624) - 4 \cdot u}}{2 \cdot 25}$

Step 4.1.5.2

Expand $(- 120 y - 624) (- 120 y - 624)$ using the FOIL Method.

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Step 4.1.5.2.1

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{- 120 y (- 120 y - 624) - 624 (- 120 y - 624) - 4 \cdot u}}{2 \cdot 25}$

Step 4.1.5.2.2

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{- 120 y (- 120 y) - 120 y \cdot - 624 - 624 (- 120 y - 624) - 4 \cdot u}}{2 \cdot 25}$

Step 4.1.5.2.3

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{- 120 y (- 120 y) - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 4.1.5.3

Simplify and combine like terms.

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Step 4.1.5.3.1

Simplify each term.

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Step 4.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{120 y + 624 \pm \sqrt{- 120 \cdot (- 120 y \cdot y) - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 4.1.5.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 4.1.5.3.1.2.1

Move $y$ .

$x = \frac{120 y + 624 \pm \sqrt{- 120 \cdot (- 120 (y \cdot y)) - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 4.1.5.3.1.2.2

Multiply $y$ by $y$ .

$x = \frac{120 y + 624 \pm \sqrt{- 120 \cdot (- 120 y^{2}) - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 4.1.5.3.1.3

Multiply $- 120$ by $- 120$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 4.1.5.3.1.4

Multiply $- 624$ by $- 120$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 74880 y - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 4.1.5.3.1.5

Multiply $- 120$ by $- 624$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 74880 y + 74880 y - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 4.1.5.3.1.6

Multiply $- 624$ by $- 624$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 74880 y + 74880 y + 389376 - 4 \cdot u}}{2 \cdot 25}$

Step 4.1.5.3.2

Add $74880 y$ and $74880 y$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 149760 y + 389376 - 4 \cdot u}}{2 \cdot 25}$

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 149760 y + 389376 - 4 u}}{2 \cdot 25}$

Step 4.1.6

Factor $4$ out of $14400 y^{2} + 149760 y + 389376 - 4 u$ .

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Step 4.1.6.1

Factor $4$ out of $14400 y^{2}$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2}) + 149760 y + 389376 - 4 u}}{2 \cdot 25}$

Step 4.1.6.2

Factor $4$ out of $149760 y$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2}) + 4 (37440 y) + 389376 - 4 u}}{2 \cdot 25}$

Step 4.1.6.3

Factor $4$ out of $389376$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2}) + 4 (37440 y) + 4 (97344) - 4 u}}{2 \cdot 25}$

Step 4.1.6.4

Factor $4$ out of $- 4 u$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2}) + 4 (37440 y) + 4 (97344) + 4 (- u)}}{2 \cdot 25}$

Step 4.1.6.5

Factor $4$ out of $4 (3600 y^{2}) + 4 (37440 y)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y) + 4 (97344) + 4 (- u)}}{2 \cdot 25}$

Step 4.1.6.6

Factor $4$ out of $4 (3600 y^{2} + 37440 y) + 4 (97344)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344) + 4 (- u)}}{2 \cdot 25}$

Step 4.1.6.7

Factor $4$ out of $4 (3600 y^{2} + 37440 y + 97344) + 4 (- u)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - u)}}{2 \cdot 25}$

Step 4.1.7

Replace all occurrences of $u$ with $25 \cdot (144 y^{2} - 260 y + 676)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (25 \cdot (144 y^{2} - 260 y + 676)))}}{2 \cdot 25}$

Step 4.1.8

Simplify.

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Step 4.1.8.1

Simplify each term.

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Step 4.1.8.1.1

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (25 (144 y^{2}) + 25 (- 260 y) + 25 \cdot 676))}}{2 \cdot 25}$

Step 4.1.8.1.2

Simplify.

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Step 4.1.8.1.2.1

Multiply $144$ by $25$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (3600 y^{2} + 25 (- 260 y) + 25 \cdot 676))}}{2 \cdot 25}$

Step 4.1.8.1.2.2

Multiply $- 260$ by $25$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (3600 y^{2} - 6500 y + 25 \cdot 676))}}{2 \cdot 25}$

Step 4.1.8.1.2.3

Multiply $25$ by $676$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (3600 y^{2} - 6500 y + 16900))}}{2 \cdot 25}$

Step 4.1.8.1.3

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (3600 y^{2}) - (- 6500 y) - 1 \cdot 16900)}}{2 \cdot 25}$

Step 4.1.8.1.4

Simplify.

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Step 4.1.8.1.4.1

Multiply $3600$ by $- 1$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - 3600 y^{2} - (- 6500 y) - 1 \cdot 16900)}}{2 \cdot 25}$

Step 4.1.8.1.4.2

Multiply $- 6500$ by $- 1$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - 3600 y^{2} + 6500 y - 1 \cdot 16900)}}{2 \cdot 25}$

Step 4.1.8.1.4.3

Multiply $- 1$ by $16900$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - 3600 y^{2} + 6500 y - 16900)}}{2 \cdot 25}$

Step 4.1.8.2

Combine the opposite terms in $3600 y^{2} + 37440 y + 97344 - 3600 y^{2} + 6500 y - 16900$ .

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Step 4.1.8.2.1

Subtract $3600 y^{2}$ from $3600 y^{2}$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (37440 y + 97344 + 0 + 6500 y - 16900)}}{2 \cdot 25}$

Step 4.1.8.2.2

Add $37440 y + 97344$ and $0$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (37440 y + 97344 + 6500 y - 16900)}}{2 \cdot 25}$

Step 4.1.8.3

Add $37440 y$ and $6500 y$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (43940 y + 97344 - 16900)}}{2 \cdot 25}$

Step 4.1.8.4

Subtract $16900$ from $97344$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (43940 y + 80444)}}{2 \cdot 25}$

Step 4.1.9

Factor $676$ out of $43940 y + 80444$ .

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Step 4.1.9.1

Factor $676$ out of $43940 y$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (676 (65 y) + 80444)}}{2 \cdot 25}$

Step 4.1.9.2

Factor $676$ out of $80444$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (676 (65 y) + 676 (119))}}{2 \cdot 25}$

Step 4.1.9.3

Factor $676$ out of $676 (65 y) + 676 (119)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (676 (65 y + 119))}}{2 \cdot 25}$

$x = \frac{120 y + 624 \pm \sqrt{4 \cdot (676 (65 y + 119))}}{2 \cdot 25}$

Step 4.1.10

Multiply $4$ by $676$ .

$x = \frac{120 y + 624 \pm \sqrt{2704 (65 y + 119)}}{2 \cdot 25}$

Step 4.1.11

Rewrite $2704$ as $52^{2}$ .

$x = \frac{120 y + 624 \pm \sqrt{52^{2} (65 y + 119)}}{2 \cdot 25}$

Step 4.1.12

Pull terms out from under the radical.

$x = \frac{120 y + 624 \pm 52 \sqrt{65 y + 119}}{2 \cdot 25}$

Step 4.2

Multiply $2$ by $25$ .

$x = \frac{120 y + 624 \pm 52 \sqrt{65 y + 119}}{50}$

Step 4.3

Change the $\pm$ to $+$ .

$x = \frac{120 y + 624 + 52 \sqrt{65 y + 119}}{50}$

Step 4.4

Cancel the common factor of $120 y + 624 + 52 \sqrt{65 y + 119}$ and $50$ .

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Step 4.4.1

Factor $2$ out of $120 y$ .

$x = \frac{2 (60 y) + 624 + 52 \sqrt{65 y + 119}}{50}$

Step 4.4.2

Factor $2$ out of $624$ .

$x = \frac{2 (60 y) + 2 (312) + 52 \sqrt{65 y + 119}}{50}$

Step 4.4.3

Factor $2$ out of $2 (60 y) + 2 (312)$ .

$x = \frac{2 (60 y + 312) + 52 \sqrt{65 y + 119}}{50}$

Step 4.4.4

Factor $2$ out of $52 \sqrt{65 y + 119}$ .

$x = \frac{2 (60 y + 312) + 2 (26 \sqrt{65 y + 119})}{50}$

Step 4.4.5

Factor $2$ out of $2 (60 y + 312) + 2 (26 \sqrt{65 y + 119})$ .

$x = \frac{2 (60 y + 312 + 26 \sqrt{65 y + 119})}{50}$

Step 4.4.6

Cancel the common factors.

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Step 4.4.6.1

Factor $2$ out of $50$ .

$x = \frac{2 (60 y + 312 + 26 \sqrt{65 y + 119})}{2 (25)}$

Step 4.4.6.2

Cancel the common factor.

$x = \frac{2 (60 y + 312 + 26 \sqrt{65 y + 119})}{2 \cdot 25}$

Step 4.4.6.3

Rewrite the expression.

$x = \frac{60 y + 312 + 26 \sqrt{65 y + 119}}{25}$

Step 4.5

Factor $2$ out of $60 y + 312 + 26 \sqrt{65 y + 119}$ .

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Step 4.5.1

Factor $2$ out of $60 y$ .

$x = \frac{2 (30 y) + 312 + 26 \sqrt{65 y + 119}}{25}$

Step 4.5.2

Factor $2$ out of $312$ .

$x = \frac{2 (30 y) + 2 (156) + 26 \sqrt{65 y + 119}}{25}$

Step 4.5.3

Factor $2$ out of $26 \sqrt{65 y + 119}$ .

$x = \frac{2 (30 y) + 2 (156) + 2 (13 \sqrt{65 y + 119})}{25}$

Step 4.5.4

Factor $2$ out of $2 (30 y) + 2 (156)$ .

$x = \frac{2 (30 y + 156) + 2 (13 \sqrt{65 y + 119})}{25}$

Step 4.5.5

Factor $2$ out of $2 (30 y + 156) + 2 (13 \sqrt{65 y + 119})$ .

$x = \frac{2 (30 y + 156 + 13 \sqrt{65 y + 119})}{25}$

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

Apply the distributive property.

$x = \frac{- (- 120 y) + 624 \pm \sqrt{{(- 120 y - 624)}^{2} - 4 \cdot 25 \cdot (144 y^{2} - 260 y + 676)}}{2 \cdot 25}$

Step 5.1.2

Multiply $- 120$ by $- 1$ .

$x = \frac{120 y + 624 \pm \sqrt{{(- 120 y - 624)}^{2} - 4 \cdot 25 \cdot (144 y^{2} - 260 y + 676)}}{2 \cdot 25}$

Step 5.1.3

Multiply $- 1$ by $- 624$ .

$x = \frac{120 y + 624 \pm \sqrt{{(- 120 y - 624)}^{2} - 4 \cdot 25 \cdot (144 y^{2} - 260 y + 676)}}{2 \cdot 25}$

Step 5.1.4

Add parentheses.

$x = \frac{120 y + 624 \pm \sqrt{{(- 120 y - 624)}^{2} - 4 \cdot (25 \cdot (144 y^{2} - 260 y + 676))}}{2 \cdot 25}$

Step 5.1.5

Let $u = 25 \cdot (144 y^{2} - 260 y + 676)$ . Substitute $u$ for all occurrences of $25 \cdot (144 y^{2} - 260 y + 676)$ .

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Step 5.1.5.1

Rewrite ${(- 120 y - 624)}^{2}$ as $(- 120 y - 624) (- 120 y - 624)$ .

$x = \frac{120 y + 624 \pm \sqrt{(- 120 y - 624) (- 120 y - 624) - 4 \cdot u}}{2 \cdot 25}$

Step 5.1.5.2

Expand $(- 120 y - 624) (- 120 y - 624)$ using the FOIL Method.

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Step 5.1.5.2.1

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{- 120 y (- 120 y - 624) - 624 (- 120 y - 624) - 4 \cdot u}}{2 \cdot 25}$

Step 5.1.5.2.2

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{- 120 y (- 120 y) - 120 y \cdot - 624 - 624 (- 120 y - 624) - 4 \cdot u}}{2 \cdot 25}$

Step 5.1.5.2.3

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{- 120 y (- 120 y) - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 5.1.5.3

Simplify and combine like terms.

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Step 5.1.5.3.1

Simplify each term.

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Step 5.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{120 y + 624 \pm \sqrt{- 120 \cdot (- 120 y \cdot y) - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 5.1.5.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 5.1.5.3.1.2.1

Move $y$ .

$x = \frac{120 y + 624 \pm \sqrt{- 120 \cdot (- 120 (y \cdot y)) - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 5.1.5.3.1.2.2

Multiply $y$ by $y$ .

$x = \frac{120 y + 624 \pm \sqrt{- 120 \cdot (- 120 y^{2}) - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 5.1.5.3.1.3

Multiply $- 120$ by $- 120$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} - 120 y \cdot - 624 - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 5.1.5.3.1.4

Multiply $- 624$ by $- 120$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 74880 y - 624 (- 120 y) - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 5.1.5.3.1.5

Multiply $- 120$ by $- 624$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 74880 y + 74880 y - 624 \cdot - 624 - 4 \cdot u}}{2 \cdot 25}$

Step 5.1.5.3.1.6

Multiply $- 624$ by $- 624$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 74880 y + 74880 y + 389376 - 4 \cdot u}}{2 \cdot 25}$

Step 5.1.5.3.2

Add $74880 y$ and $74880 y$ .

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 149760 y + 389376 - 4 \cdot u}}{2 \cdot 25}$

$x = \frac{120 y + 624 \pm \sqrt{14400 y^{2} + 149760 y + 389376 - 4 u}}{2 \cdot 25}$

Step 5.1.6

Factor $4$ out of $14400 y^{2} + 149760 y + 389376 - 4 u$ .

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Step 5.1.6.1

Factor $4$ out of $14400 y^{2}$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2}) + 149760 y + 389376 - 4 u}}{2 \cdot 25}$

Step 5.1.6.2

Factor $4$ out of $149760 y$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2}) + 4 (37440 y) + 389376 - 4 u}}{2 \cdot 25}$

Step 5.1.6.3

Factor $4$ out of $389376$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2}) + 4 (37440 y) + 4 (97344) - 4 u}}{2 \cdot 25}$

Step 5.1.6.4

Factor $4$ out of $- 4 u$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2}) + 4 (37440 y) + 4 (97344) + 4 (- u)}}{2 \cdot 25}$

Step 5.1.6.5

Factor $4$ out of $4 (3600 y^{2}) + 4 (37440 y)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y) + 4 (97344) + 4 (- u)}}{2 \cdot 25}$

Step 5.1.6.6

Factor $4$ out of $4 (3600 y^{2} + 37440 y) + 4 (97344)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344) + 4 (- u)}}{2 \cdot 25}$

Step 5.1.6.7

Factor $4$ out of $4 (3600 y^{2} + 37440 y + 97344) + 4 (- u)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - u)}}{2 \cdot 25}$

Step 5.1.7

Replace all occurrences of $u$ with $25 \cdot (144 y^{2} - 260 y + 676)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (25 \cdot (144 y^{2} - 260 y + 676)))}}{2 \cdot 25}$

Step 5.1.8

Simplify.

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Step 5.1.8.1

Simplify each term.

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Step 5.1.8.1.1

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (25 (144 y^{2}) + 25 (- 260 y) + 25 \cdot 676))}}{2 \cdot 25}$

Step 5.1.8.1.2

Simplify.

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Step 5.1.8.1.2.1

Multiply $144$ by $25$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (3600 y^{2} + 25 (- 260 y) + 25 \cdot 676))}}{2 \cdot 25}$

Step 5.1.8.1.2.2

Multiply $- 260$ by $25$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (3600 y^{2} - 6500 y + 25 \cdot 676))}}{2 \cdot 25}$

Step 5.1.8.1.2.3

Multiply $25$ by $676$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (3600 y^{2} - 6500 y + 16900))}}{2 \cdot 25}$

Step 5.1.8.1.3

Apply the distributive property.

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - (3600 y^{2}) - (- 6500 y) - 1 \cdot 16900)}}{2 \cdot 25}$

Step 5.1.8.1.4

Simplify.

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Step 5.1.8.1.4.1

Multiply $3600$ by $- 1$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - 3600 y^{2} - (- 6500 y) - 1 \cdot 16900)}}{2 \cdot 25}$

Step 5.1.8.1.4.2

Multiply $- 6500$ by $- 1$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - 3600 y^{2} + 6500 y - 1 \cdot 16900)}}{2 \cdot 25}$

Step 5.1.8.1.4.3

Multiply $- 1$ by $16900$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (3600 y^{2} + 37440 y + 97344 - 3600 y^{2} + 6500 y - 16900)}}{2 \cdot 25}$

Step 5.1.8.2

Combine the opposite terms in $3600 y^{2} + 37440 y + 97344 - 3600 y^{2} + 6500 y - 16900$ .

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Step 5.1.8.2.1

Subtract $3600 y^{2}$ from $3600 y^{2}$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (37440 y + 97344 + 0 + 6500 y - 16900)}}{2 \cdot 25}$

Step 5.1.8.2.2

Add $37440 y + 97344$ and $0$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (37440 y + 97344 + 6500 y - 16900)}}{2 \cdot 25}$

Step 5.1.8.3

Add $37440 y$ and $6500 y$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (43940 y + 97344 - 16900)}}{2 \cdot 25}$

Step 5.1.8.4

Subtract $16900$ from $97344$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (43940 y + 80444)}}{2 \cdot 25}$

Step 5.1.9

Factor $676$ out of $43940 y + 80444$ .

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Step 5.1.9.1

Factor $676$ out of $43940 y$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (676 (65 y) + 80444)}}{2 \cdot 25}$

Step 5.1.9.2

Factor $676$ out of $80444$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (676 (65 y) + 676 (119))}}{2 \cdot 25}$

Step 5.1.9.3

Factor $676$ out of $676 (65 y) + 676 (119)$ .

$x = \frac{120 y + 624 \pm \sqrt{4 (676 (65 y + 119))}}{2 \cdot 25}$

$x = \frac{120 y + 624 \pm \sqrt{4 \cdot (676 (65 y + 119))}}{2 \cdot 25}$

Step 5.1.10

Multiply $4$ by $676$ .

$x = \frac{120 y + 624 \pm \sqrt{2704 (65 y + 119)}}{2 \cdot 25}$

Step 5.1.11

Rewrite $2704$ as $52^{2}$ .

$x = \frac{120 y + 624 \pm \sqrt{52^{2} (65 y + 119)}}{2 \cdot 25}$

Step 5.1.12

Pull terms out from under the radical.

$x = \frac{120 y + 624 \pm 52 \sqrt{65 y + 119}}{2 \cdot 25}$

Step 5.2

Multiply $2$ by $25$ .

$x = \frac{120 y + 624 \pm 52 \sqrt{65 y + 119}}{50}$

Step 5.3

Change the $\pm$ to $-$ .

$x = \frac{120 y + 624 - 52 \sqrt{65 y + 119}}{50}$

Step 5.4

Cancel the common factor of $120 y + 624 - 52 \sqrt{65 y + 119}$ and $50$ .

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Step 5.4.1

Factor $2$ out of $120 y$ .

$x = \frac{2 (60 y) + 624 - 52 \sqrt{65 y + 119}}{50}$

Step 5.4.2

Factor $2$ out of $624$ .

$x = \frac{2 (60 y) + 2 (312) - 52 \sqrt{65 y + 119}}{50}$

Step 5.4.3

Factor $2$ out of $2 (60 y) + 2 (312)$ .

$x = \frac{2 (60 y + 312) - 52 \sqrt{65 y + 119}}{50}$

Step 5.4.4

Factor $2$ out of $- 52 \sqrt{65 y + 119}$ .

$x = \frac{2 (60 y + 312) + 2 (- 26 \sqrt{65 y + 119})}{50}$

Step 5.4.5

Factor $2$ out of $2 (60 y + 312) + 2 (- 26 \sqrt{65 y + 119})$ .

$x = \frac{2 (60 y + 312 - 26 \sqrt{65 y + 119})}{50}$

Step 5.4.6

Cancel the common factors.

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Step 5.4.6.1

Factor $2$ out of $50$ .

$x = \frac{2 (60 y + 312 - 26 \sqrt{65 y + 119})}{2 (25)}$

Step 5.4.6.2

Cancel the common factor.

$x = \frac{2 (60 y + 312 - 26 \sqrt{65 y + 119})}{2 \cdot 25}$

Step 5.4.6.3

Rewrite the expression.

$x = \frac{60 y + 312 - 26 \sqrt{65 y + 119}}{25}$

Step 5.5

Factor $2$ out of $60 y + 312 - 26 \sqrt{65 y + 119}$ .

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Step 5.5.1

Factor $2$ out of $60 y$ .

$x = \frac{2 (30 y) + 312 - 26 \sqrt{65 y + 119}}{25}$

Step 5.5.2

Factor $2$ out of $312$ .

$x = \frac{2 (30 y) + 2 (156) - 26 \sqrt{65 y + 119}}{25}$

Step 5.5.3

Factor $2$ out of $- 26 \sqrt{65 y + 119}$ .

$x = \frac{2 (30 y) + 2 (156) + 2 (- 13 \sqrt{65 y + 119})}{25}$

Step 5.5.4

Factor $2$ out of $2 (30 y) + 2 (156)$ .

$x = \frac{2 (30 y + 156) + 2 (- 13 \sqrt{65 y + 119})}{25}$

Step 5.5.5

Factor $2$ out of $2 (30 y + 156) + 2 (- 13 \sqrt{65 y + 119})$ .

$x = \frac{2 (30 y + 156 - 13 \sqrt{65 y + 119})}{25}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{2 (30 y + 156 + 13 \sqrt{65 y + 119})}{25}$

$x = \frac{2 (30 y + 156 - 13 \sqrt{65 y + 119})}{25}$