Trigonometry Examples

$\sqrt{5 x^{2}} - 10 x + 52$

Step 1

Interchange the variables.

$x = \sqrt{5 y^{2}} - 10 y + 52$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\sqrt{5 y^{2}} - 10 y + 52 = x$ .

$\sqrt{5 y^{2}} - 10 y + 52 = x$

Step 2.2

Move all terms not containing $\sqrt{5 y^{2}}$ to the right side of the equation.

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Step 2.2.1

Add $10 y$ to both sides of the equation.

$\sqrt{5 y^{2}} + 52 = x + 10 y$

Step 2.2.2

Subtract $52$ from both sides of the equation.

$\sqrt{5 y^{2}} = x + 10 y - 52$

Step 2.3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{5 y^{2}}}^{2} = {(x + 10 y - 52)}^{2}$

Step 2.4

Simplify each side of the equation.

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Step 2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 y^{2}}$ as ${(5 y^{2})}^{\frac{1}{2}}$ .

${({(5 y^{2})}^{\frac{1}{2}})}^{2} = {(x + 10 y - 52)}^{2}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Simplify ${({(5 y^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.4.2.1.1

Multiply the exponents in ${({(5 y^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.4.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(5 y^{2})}^{\frac{1}{2} \cdot 2} = {(x + 10 y - 52)}^{2}$

Step 2.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.4.2.1.1.2.1

Cancel the common factor.

${(5 y^{2})}^{\frac{1}{2} \cdot 2} = {(x + 10 y - 52)}^{2}$

Step 2.4.2.1.1.2.2

Rewrite the expression.

${(5 y^{2})}^{1} = {(x + 10 y - 52)}^{2}$

Step 2.4.2.1.2

Simplify.

$5 y^{2} = {(x + 10 y - 52)}^{2}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify ${(x + 10 y - 52)}^{2}$ .

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Step 2.4.3.1.1

Rewrite ${(x + 10 y - 52)}^{2}$ as $(x + 10 y - 52) (x + 10 y - 52)$ .

$5 y^{2} = (x + 10 y - 52) (x + 10 y - 52)$

Step 2.4.3.1.2

Expand $(x + 10 y - 52) (x + 10 y - 52)$ by multiplying each term in the first expression by each term in the second expression.

$5 y^{2} = x \cdot x + x (10 y) + x \cdot - 52 + 10 y x + 10 y (10 y) + 10 y \cdot - 52 - 52 x - 52 (10 y) - 52 \cdot - 52$

Step 2.4.3.1.3

Simplify each term.

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Step 2.4.3.1.3.1

Multiply $x$ by $x$ .

$5 y^{2} = x^{2} + x (10 y) + x \cdot - 52 + 10 y x + 10 y (10 y) + 10 y \cdot - 52 - 52 x - 52 (10 y) - 52 \cdot - 52$

Step 2.4.3.1.3.2

Rewrite using the commutative property of multiplication.

$5 y^{2} = x^{2} + 10 x y + x \cdot - 52 + 10 y x + 10 y (10 y) + 10 y \cdot - 52 - 52 x - 52 (10 y) - 52 \cdot - 52$

Step 2.4.3.1.3.3

Move $- 52$ to the left of $x$ .

$5 y^{2} = x^{2} + 10 x y - 52 \cdot x + 10 y x + 10 y (10 y) + 10 y \cdot - 52 - 52 x - 52 (10 y) - 52 \cdot - 52$

Step 2.4.3.1.3.4

Rewrite using the commutative property of multiplication.

$5 y^{2} = x^{2} + 10 x y - 52 x + 10 y x + 10 \cdot 10 y \cdot y + 10 y \cdot - 52 - 52 x - 52 (10 y) - 52 \cdot - 52$

Step 2.4.3.1.3.5

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

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Step 2.4.3.1.3.5.1

Move $y$ .

$5 y^{2} = x^{2} + 10 x y - 52 x + 10 y x + 10 \cdot 10 (y \cdot y) + 10 y \cdot - 52 - 52 x - 52 (10 y) - 52 \cdot - 52$

Step 2.4.3.1.3.5.2

Multiply $y$ by $y$ .

$5 y^{2} = x^{2} + 10 x y - 52 x + 10 y x + 10 \cdot 10 y^{2} + 10 y \cdot - 52 - 52 x - 52 (10 y) - 52 \cdot - 52$

Step 2.4.3.1.3.6

Multiply $10$ by $10$ .

$5 y^{2} = x^{2} + 10 x y - 52 x + 10 y x + 100 y^{2} + 10 y \cdot - 52 - 52 x - 52 (10 y) - 52 \cdot - 52$

Step 2.4.3.1.3.7

Multiply $- 52$ by $10$ .

$5 y^{2} = x^{2} + 10 x y - 52 x + 10 y x + 100 y^{2} - 520 y - 52 x - 52 (10 y) - 52 \cdot - 52$

Step 2.4.3.1.3.8

Multiply $10$ by $- 52$ .

$5 y^{2} = x^{2} + 10 x y - 52 x + 10 y x + 100 y^{2} - 520 y - 52 x - 520 y - 52 \cdot - 52$

Step 2.4.3.1.3.9

Multiply $- 52$ by $- 52$ .

$5 y^{2} = x^{2} + 10 x y - 52 x + 10 y x + 100 y^{2} - 520 y - 52 x - 520 y + 2704$

Step 2.4.3.1.4

Add $10 x y$ and $10 y x$ .

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Step 2.4.3.1.4.1

Move $y$ .

$5 y^{2} = x^{2} + 10 x y + 10 x y - 52 x + 100 y^{2} - 520 y - 52 x - 520 y + 2704$

Step 2.4.3.1.4.2

Add $10 x y$ and $10 x y$ .

$5 y^{2} = x^{2} + 20 x y - 52 x + 100 y^{2} - 520 y - 52 x - 520 y + 2704$

Step 2.4.3.1.5

Subtract $52 x$ from $- 52 x$ .

$5 y^{2} = x^{2} + 20 x y - 104 x + 100 y^{2} - 520 y - 520 y + 2704$

Step 2.4.3.1.6

Subtract $520 y$ from $- 520 y$ .

$5 y^{2} = x^{2} + 20 x y - 104 x + 100 y^{2} - 1040 y + 2704$

Step 2.5

Solve for $y$ .

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Step 2.5.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 20 x y - 104 x + 100 y^{2} - 1040 y + 2704 = 5 y^{2}$

Step 2.5.2

Move all terms containing $y$ to the left side of the equation.

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Step 2.5.2.1

Subtract $5 y^{2}$ from both sides of the equation.

$x^{2} + 20 x y - 104 x + 100 y^{2} - 1040 y + 2704 - 5 y^{2} = 0$

Step 2.5.2.2

Subtract $5 y^{2}$ from $100 y^{2}$ .

$x^{2} + 20 x y - 104 x + 95 y^{2} - 1040 y + 2704 = 0$

Step 2.5.3

Use the quadratic formula to find the solutions.

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

Step 2.5.4

Substitute the values $a = 95$ , $b = 20 x - 1040$ , and $c = x^{2} - 104 x + 2704$ into the quadratic formula and solve for $y$ .

$\frac{- (20 x - 1040) \pm \sqrt{{(20 x - 1040)}^{2} - 4 \cdot (95 \cdot (x^{2} - 104 x + 2704))}}{2 \cdot 95}$

Step 2.5.5

Simplify.

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Step 2.5.5.1

Simplify the numerator.

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Step 2.5.5.1.1

Apply the distributive property.

$y = \frac{- (20 x) + 1040 \pm \sqrt{{(20 x - 1040)}^{2} - 4 \cdot 95 \cdot (x^{2} - 104 x + 2704)}}{2 \cdot 95}$

Step 2.5.5.1.2

Multiply $20$ by $- 1$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{{(20 x - 1040)}^{2} - 4 \cdot 95 \cdot (x^{2} - 104 x + 2704)}}{2 \cdot 95}$

Step 2.5.5.1.3

Multiply $- 1$ by $- 1040$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{{(20 x - 1040)}^{2} - 4 \cdot 95 \cdot (x^{2} - 104 x + 2704)}}{2 \cdot 95}$

Step 2.5.5.1.4

Add parentheses.

$y = \frac{- 20 x + 1040 \pm \sqrt{{(20 x - 1040)}^{2} - 4 \cdot (95 \cdot (x^{2} - 104 x + 2704))}}{2 \cdot 95}$

Step 2.5.5.1.5

Let $u = 95 \cdot (x^{2} - 104 x + 2704)$ . Substitute $u$ for all occurrences of $95 \cdot (x^{2} - 104 x + 2704)$ .

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Step 2.5.5.1.5.1

Rewrite ${(20 x - 1040)}^{2}$ as $(20 x - 1040) (20 x - 1040)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{(20 x - 1040) (20 x - 1040) - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.5.1.5.2

Expand $(20 x - 1040) (20 x - 1040)$ using the FOIL Method.

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Step 2.5.5.1.5.2.1

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{20 x (20 x - 1040) - 1040 (20 x - 1040) - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.5.1.5.2.2

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{20 x (20 x) + 20 x \cdot - 1040 - 1040 (20 x - 1040) - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.5.1.5.2.3

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{20 x (20 x) + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.5.1.5.3

Simplify and combine like terms.

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Step 2.5.5.1.5.3.1

Simplify each term.

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Step 2.5.5.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{- 20 x + 1040 \pm \sqrt{20 \cdot (20 x \cdot x) + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.5.1.5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.5.5.1.5.3.1.2.1

Move $x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{20 \cdot (20 (x \cdot x)) + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.5.1.5.3.1.2.2

Multiply $x$ by $x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{20 \cdot (20 x^{2}) + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.5.1.5.3.1.3

Multiply $20$ by $20$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.5.1.5.3.1.4

Multiply $- 1040$ by $20$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 20800 x - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.5.1.5.3.1.5

Multiply $20$ by $- 1040$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 20800 x - 20800 x - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.5.1.5.3.1.6

Multiply $- 1040$ by $- 1040$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 20800 x - 20800 x + 1081600 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.5.1.5.3.2

Subtract $20800 x$ from $- 20800 x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 41600 x + 1081600 - 4 \cdot u}}{2 \cdot 95}$

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 41600 x + 1081600 - 4 u}}{2 \cdot 95}$

Step 2.5.5.1.6

Factor $4$ out of $400 x^{2} - 41600 x + 1081600 - 4 u$ .

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Step 2.5.5.1.6.1

Factor $4$ out of $400 x^{2}$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2}) - 41600 x + 1081600 - 4 u}}{2 \cdot 95}$

Step 2.5.5.1.6.2

Factor $4$ out of $- 41600 x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2}) + 4 (- 10400 x) + 1081600 - 4 u}}{2 \cdot 95}$

Step 2.5.5.1.6.3

Factor $4$ out of $1081600$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2}) + 4 (- 10400 x) + 4 (270400) - 4 u}}{2 \cdot 95}$

Step 2.5.5.1.6.4

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

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2}) + 4 (- 10400 x) + 4 (270400) + 4 (- u)}}{2 \cdot 95}$

Step 2.5.5.1.6.5

Factor $4$ out of $4 (100 x^{2}) + 4 (- 10400 x)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x) + 4 (270400) + 4 (- u)}}{2 \cdot 95}$

Step 2.5.5.1.6.6

Factor $4$ out of $4 (100 x^{2} - 10400 x) + 4 (270400)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400) + 4 (- u)}}{2 \cdot 95}$

Step 2.5.5.1.6.7

Factor $4$ out of $4 (100 x^{2} - 10400 x + 270400) + 4 (- u)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - u)}}{2 \cdot 95}$

Step 2.5.5.1.7

Replace all occurrences of $u$ with $95 \cdot (x^{2} - 104 x + 2704)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 \cdot (x^{2} - 104 x + 2704)))}}{2 \cdot 95}$

Step 2.5.5.1.8

Simplify.

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Step 2.5.5.1.8.1

Simplify each term.

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Step 2.5.5.1.8.1.1

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 x^{2} + 95 (- 104 x) + 95 \cdot 2704))}}{2 \cdot 95}$

Step 2.5.5.1.8.1.2

Simplify.

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Step 2.5.5.1.8.1.2.1

Multiply $- 104$ by $95$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 x^{2} - 9880 x + 95 \cdot 2704))}}{2 \cdot 95}$

Step 2.5.5.1.8.1.2.2

Multiply $95$ by $2704$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 x^{2} - 9880 x + 256880))}}{2 \cdot 95}$

Step 2.5.5.1.8.1.3

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 x^{2}) - (- 9880 x) - 1 \cdot 256880)}}{2 \cdot 95}$

Step 2.5.5.1.8.1.4

Simplify.

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Step 2.5.5.1.8.1.4.1

Multiply $95$ by $- 1$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - 95 x^{2} - (- 9880 x) - 1 \cdot 256880)}}{2 \cdot 95}$

Step 2.5.5.1.8.1.4.2

Multiply $- 9880$ by $- 1$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - 95 x^{2} + 9880 x - 1 \cdot 256880)}}{2 \cdot 95}$

Step 2.5.5.1.8.1.4.3

Multiply $- 1$ by $256880$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - 95 x^{2} + 9880 x - 256880)}}{2 \cdot 95}$

Step 2.5.5.1.8.2

Subtract $95 x^{2}$ from $100 x^{2}$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 x^{2} - 10400 x + 270400 + 9880 x - 256880)}}{2 \cdot 95}$

Step 2.5.5.1.8.3

Add $- 10400 x$ and $9880 x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 x^{2} - 520 x + 270400 - 256880)}}{2 \cdot 95}$

Step 2.5.5.1.8.4

Subtract $256880$ from $270400$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 x^{2} - 520 x + 13520)}}{2 \cdot 95}$

Step 2.5.5.1.9

Factor $5$ out of $5 x^{2} - 520 x + 13520$ .

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Step 2.5.5.1.9.1

Factor $5$ out of $5 x^{2}$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2}) - 520 x + 13520)}}{2 \cdot 95}$

Step 2.5.5.1.9.2

Factor $5$ out of $- 520 x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2}) + 5 (- 104 x) + 13520)}}{2 \cdot 95}$

Step 2.5.5.1.9.3

Factor $5$ out of $13520$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 x^{2} + 5 (- 104 x) + 5 \cdot 2704)}}{2 \cdot 95}$

Step 2.5.5.1.9.4

Factor $5$ out of $5 x^{2} + 5 (- 104 x)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2} - 104 x) + 5 \cdot 2704)}}{2 \cdot 95}$

Step 2.5.5.1.9.5

Factor $5$ out of $5 (x^{2} - 104 x) + 5 \cdot 2704$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2} - 104 x + 2704))}}{2 \cdot 95}$

Step 2.5.5.1.10

Factor using the perfect square rule.

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Step 2.5.5.1.10.1

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

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2} - 104 x + 52^{2}))}}{2 \cdot 95}$

Step 2.5.5.1.10.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$104 x = 2 \cdot x \cdot 52$

Step 2.5.5.1.10.3

Rewrite the polynomial.

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2} - 2 \cdot x \cdot 52 + 52^{2}))}}{2 \cdot 95}$

Step 2.5.5.1.10.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 52$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 {(x - 52)}^{2})}}{2 \cdot 95}$

$y = \frac{- 20 x + 1040 \pm \sqrt{4 \cdot (5 {(x - 52)}^{2})}}{2 \cdot 95}$

Step 2.5.5.1.11

Multiply $4$ by $5$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{20 {(x - 52)}^{2}}}{2 \cdot 95}$

Step 2.5.5.1.12

Rewrite $20 {(x - 52)}^{2}$ as ${(2 (x - 52))}^{2} \cdot 5$ .

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Step 2.5.5.1.12.1

Factor $4$ out of $20$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5) {(x - 52)}^{2}}}{2 \cdot 95}$

Step 2.5.5.1.12.2

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

$y = \frac{- 20 x + 1040 \pm \sqrt{2^{2} \cdot (5 {(x - 52)}^{2})}}{2 \cdot 95}$

Step 2.5.5.1.12.3

Move $5$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{2^{2} {(x - 52)}^{2} \cdot 5}}{2 \cdot 95}$

Step 2.5.5.1.12.4

Rewrite $2^{2} {(x - 52)}^{2}$ as ${(2 (x - 52))}^{2}$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{{(2 (x - 52))}^{2} \cdot 5}}{2 \cdot 95}$

Step 2.5.5.1.13

Pull terms out from under the radical.

$y = \frac{- 20 x + 1040 \pm 2 (x - 52) \sqrt{5}}{2 \cdot 95}$

Step 2.5.5.1.14

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm (2 x + 2 \cdot - 52) \sqrt{5}}{2 \cdot 95}$

Step 2.5.5.1.15

Multiply $2$ by $- 52$ .

$y = \frac{- 20 x + 1040 \pm (2 x - 104) \sqrt{5}}{2 \cdot 95}$

Step 2.5.5.1.16

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm (2 x \sqrt{5} - 104 \sqrt{5})}{2 \cdot 95}$

Step 2.5.5.2

Multiply $2$ by $95$ .

$y = \frac{- 20 x + 1040 \pm (2 x \sqrt{5} - 104 \sqrt{5})}{190}$

Step 2.5.6

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

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Step 2.5.6.1

Simplify the numerator.

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Step 2.5.6.1.1

Apply the distributive property.

$y = \frac{- (20 x) + 1040 \pm \sqrt{{(20 x - 1040)}^{2} - 4 \cdot 95 \cdot (x^{2} - 104 x + 2704)}}{2 \cdot 95}$

Step 2.5.6.1.2

Multiply $20$ by $- 1$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{{(20 x - 1040)}^{2} - 4 \cdot 95 \cdot (x^{2} - 104 x + 2704)}}{2 \cdot 95}$

Step 2.5.6.1.3

Multiply $- 1$ by $- 1040$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{{(20 x - 1040)}^{2} - 4 \cdot 95 \cdot (x^{2} - 104 x + 2704)}}{2 \cdot 95}$

Step 2.5.6.1.4

Add parentheses.

$y = \frac{- 20 x + 1040 \pm \sqrt{{(20 x - 1040)}^{2} - 4 \cdot (95 \cdot (x^{2} - 104 x + 2704))}}{2 \cdot 95}$

Step 2.5.6.1.5

Let $u = 95 \cdot (x^{2} - 104 x + 2704)$ . Substitute $u$ for all occurrences of $95 \cdot (x^{2} - 104 x + 2704)$ .

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Step 2.5.6.1.5.1

Rewrite ${(20 x - 1040)}^{2}$ as $(20 x - 1040) (20 x - 1040)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{(20 x - 1040) (20 x - 1040) - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.6.1.5.2

Expand $(20 x - 1040) (20 x - 1040)$ using the FOIL Method.

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Step 2.5.6.1.5.2.1

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{20 x (20 x - 1040) - 1040 (20 x - 1040) - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.6.1.5.2.2

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{20 x (20 x) + 20 x \cdot - 1040 - 1040 (20 x - 1040) - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.6.1.5.2.3

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{20 x (20 x) + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.6.1.5.3

Simplify and combine like terms.

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Step 2.5.6.1.5.3.1

Simplify each term.

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Step 2.5.6.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{- 20 x + 1040 \pm \sqrt{20 \cdot (20 x \cdot x) + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.6.1.5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.5.6.1.5.3.1.2.1

Move $x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{20 \cdot (20 (x \cdot x)) + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.6.1.5.3.1.2.2

Multiply $x$ by $x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{20 \cdot (20 x^{2}) + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.6.1.5.3.1.3

Multiply $20$ by $20$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.6.1.5.3.1.4

Multiply $- 1040$ by $20$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 20800 x - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.6.1.5.3.1.5

Multiply $20$ by $- 1040$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 20800 x - 20800 x - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.6.1.5.3.1.6

Multiply $- 1040$ by $- 1040$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 20800 x - 20800 x + 1081600 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.6.1.5.3.2

Subtract $20800 x$ from $- 20800 x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 41600 x + 1081600 - 4 \cdot u}}{2 \cdot 95}$

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 41600 x + 1081600 - 4 u}}{2 \cdot 95}$

Step 2.5.6.1.6

Factor $4$ out of $400 x^{2} - 41600 x + 1081600 - 4 u$ .

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Step 2.5.6.1.6.1

Factor $4$ out of $400 x^{2}$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2}) - 41600 x + 1081600 - 4 u}}{2 \cdot 95}$

Step 2.5.6.1.6.2

Factor $4$ out of $- 41600 x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2}) + 4 (- 10400 x) + 1081600 - 4 u}}{2 \cdot 95}$

Step 2.5.6.1.6.3

Factor $4$ out of $1081600$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2}) + 4 (- 10400 x) + 4 (270400) - 4 u}}{2 \cdot 95}$

Step 2.5.6.1.6.4

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

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2}) + 4 (- 10400 x) + 4 (270400) + 4 (- u)}}{2 \cdot 95}$

Step 2.5.6.1.6.5

Factor $4$ out of $4 (100 x^{2}) + 4 (- 10400 x)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x) + 4 (270400) + 4 (- u)}}{2 \cdot 95}$

Step 2.5.6.1.6.6

Factor $4$ out of $4 (100 x^{2} - 10400 x) + 4 (270400)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400) + 4 (- u)}}{2 \cdot 95}$

Step 2.5.6.1.6.7

Factor $4$ out of $4 (100 x^{2} - 10400 x + 270400) + 4 (- u)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - u)}}{2 \cdot 95}$

Step 2.5.6.1.7

Replace all occurrences of $u$ with $95 \cdot (x^{2} - 104 x + 2704)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 \cdot (x^{2} - 104 x + 2704)))}}{2 \cdot 95}$

Step 2.5.6.1.8

Simplify.

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Step 2.5.6.1.8.1

Simplify each term.

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Step 2.5.6.1.8.1.1

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 x^{2} + 95 (- 104 x) + 95 \cdot 2704))}}{2 \cdot 95}$

Step 2.5.6.1.8.1.2

Simplify.

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Step 2.5.6.1.8.1.2.1

Multiply $- 104$ by $95$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 x^{2} - 9880 x + 95 \cdot 2704))}}{2 \cdot 95}$

Step 2.5.6.1.8.1.2.2

Multiply $95$ by $2704$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 x^{2} - 9880 x + 256880))}}{2 \cdot 95}$

Step 2.5.6.1.8.1.3

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 x^{2}) - (- 9880 x) - 1 \cdot 256880)}}{2 \cdot 95}$

Step 2.5.6.1.8.1.4

Simplify.

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Step 2.5.6.1.8.1.4.1

Multiply $95$ by $- 1$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - 95 x^{2} - (- 9880 x) - 1 \cdot 256880)}}{2 \cdot 95}$

Step 2.5.6.1.8.1.4.2

Multiply $- 9880$ by $- 1$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - 95 x^{2} + 9880 x - 1 \cdot 256880)}}{2 \cdot 95}$

Step 2.5.6.1.8.1.4.3

Multiply $- 1$ by $256880$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - 95 x^{2} + 9880 x - 256880)}}{2 \cdot 95}$

Step 2.5.6.1.8.2

Subtract $95 x^{2}$ from $100 x^{2}$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 x^{2} - 10400 x + 270400 + 9880 x - 256880)}}{2 \cdot 95}$

Step 2.5.6.1.8.3

Add $- 10400 x$ and $9880 x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 x^{2} - 520 x + 270400 - 256880)}}{2 \cdot 95}$

Step 2.5.6.1.8.4

Subtract $256880$ from $270400$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 x^{2} - 520 x + 13520)}}{2 \cdot 95}$

Step 2.5.6.1.9

Factor $5$ out of $5 x^{2} - 520 x + 13520$ .

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Step 2.5.6.1.9.1

Factor $5$ out of $5 x^{2}$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2}) - 520 x + 13520)}}{2 \cdot 95}$

Step 2.5.6.1.9.2

Factor $5$ out of $- 520 x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2}) + 5 (- 104 x) + 13520)}}{2 \cdot 95}$

Step 2.5.6.1.9.3

Factor $5$ out of $13520$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 x^{2} + 5 (- 104 x) + 5 \cdot 2704)}}{2 \cdot 95}$

Step 2.5.6.1.9.4

Factor $5$ out of $5 x^{2} + 5 (- 104 x)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2} - 104 x) + 5 \cdot 2704)}}{2 \cdot 95}$

Step 2.5.6.1.9.5

Factor $5$ out of $5 (x^{2} - 104 x) + 5 \cdot 2704$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2} - 104 x + 2704))}}{2 \cdot 95}$

Step 2.5.6.1.10

Factor using the perfect square rule.

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Step 2.5.6.1.10.1

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

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2} - 104 x + 52^{2}))}}{2 \cdot 95}$

Step 2.5.6.1.10.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$104 x = 2 \cdot x \cdot 52$

Step 2.5.6.1.10.3

Rewrite the polynomial.

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2} - 2 \cdot x \cdot 52 + 52^{2}))}}{2 \cdot 95}$

Step 2.5.6.1.10.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 52$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 {(x - 52)}^{2})}}{2 \cdot 95}$

$y = \frac{- 20 x + 1040 \pm \sqrt{4 \cdot (5 {(x - 52)}^{2})}}{2 \cdot 95}$

Step 2.5.6.1.11

Multiply $4$ by $5$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{20 {(x - 52)}^{2}}}{2 \cdot 95}$

Step 2.5.6.1.12

Rewrite $20 {(x - 52)}^{2}$ as ${(2 (x - 52))}^{2} \cdot 5$ .

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Step 2.5.6.1.12.1

Factor $4$ out of $20$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5) {(x - 52)}^{2}}}{2 \cdot 95}$

Step 2.5.6.1.12.2

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

$y = \frac{- 20 x + 1040 \pm \sqrt{2^{2} \cdot (5 {(x - 52)}^{2})}}{2 \cdot 95}$

Step 2.5.6.1.12.3

Move $5$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{2^{2} {(x - 52)}^{2} \cdot 5}}{2 \cdot 95}$

Step 2.5.6.1.12.4

Rewrite $2^{2} {(x - 52)}^{2}$ as ${(2 (x - 52))}^{2}$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{{(2 (x - 52))}^{2} \cdot 5}}{2 \cdot 95}$

Step 2.5.6.1.13

Pull terms out from under the radical.

$y = \frac{- 20 x + 1040 \pm 2 (x - 52) \sqrt{5}}{2 \cdot 95}$

Step 2.5.6.1.14

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm (2 x + 2 \cdot - 52) \sqrt{5}}{2 \cdot 95}$

Step 2.5.6.1.15

Multiply $2$ by $- 52$ .

$y = \frac{- 20 x + 1040 \pm (2 x - 104) \sqrt{5}}{2 \cdot 95}$

Step 2.5.6.1.16

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm (2 x \sqrt{5} - 104 \sqrt{5})}{2 \cdot 95}$

Step 2.5.6.2

Multiply $2$ by $95$ .

$y = \frac{- 20 x + 1040 \pm (2 x \sqrt{5} - 104 \sqrt{5})}{190}$

Step 2.5.6.3

Change the $\pm$ to $+$ .

$y = \frac{- 20 x + 1040 + 2 x \sqrt{5} - 104 \sqrt{5}}{190}$

Step 2.5.6.4

Cancel the common factor of $- 20 x + 1040 + 2 x \sqrt{5} - 104 \sqrt{5}$ and $190$ .

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Step 2.5.6.4.1

Factor $2$ out of $- 20 x$ .

$y = \frac{2 (- 10 x) + 1040 + 2 x \sqrt{5} - 104 \sqrt{5}}{190}$

Step 2.5.6.4.2

Factor $2$ out of $1040$ .

$y = \frac{2 (- 10 x) + 2 \cdot 520 + 2 x \sqrt{5} - 104 \sqrt{5}}{190}$

Step 2.5.6.4.3

Factor $2$ out of $2 (- 10 x) + 2 (520)$ .

$y = \frac{2 (- 10 x + 520) + 2 x \sqrt{5} - 104 \sqrt{5}}{190}$

Step 2.5.6.4.4

Factor $2$ out of $2 x \sqrt{5}$ .

$y = \frac{2 (- 10 x + 520) + 2 (x \sqrt{5}) - 104 \sqrt{5}}{190}$

Step 2.5.6.4.5

Factor $2$ out of $- 104 \sqrt{5}$ .

$y = \frac{2 (- 10 x + 520) + 2 (x \sqrt{5}) + 2 (- 52 \sqrt{5})}{190}$

Step 2.5.6.4.6

Factor $2$ out of $2 (x \sqrt{5}) + 2 (- 52 \sqrt{5})$ .

$y = \frac{2 (- 10 x + 520) + 2 (x \sqrt{5} - 52 \sqrt{5})}{190}$

Step 2.5.6.4.7

Factor $2$ out of $2 (- 10 x + 520) + 2 (x \sqrt{5} - 52 \sqrt{5})$ .

$y = \frac{2 (- 10 x + 520 + x \sqrt{5} - 52 \sqrt{5})}{190}$

Step 2.5.6.4.8

Cancel the common factors.

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Step 2.5.6.4.8.1

Factor $2$ out of $190$ .

$y = \frac{2 (- 10 x + 520 + x \sqrt{5} - 52 \sqrt{5})}{2 (95)}$

Step 2.5.6.4.8.2

Cancel the common factor.

$y = \frac{2 (- 10 x + 520 + x \sqrt{5} - 52 \sqrt{5})}{2 \cdot 95}$

Step 2.5.6.4.8.3

Rewrite the expression.

$y = \frac{- 10 x + 520 + x \sqrt{5} - 52 \sqrt{5}}{95}$

Step 2.5.6.5

Reorder terms.

$y = \frac{x \sqrt{5} - 10 x + 520 - 52 \sqrt{5}}{95}$

Step 2.5.7

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

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Step 2.5.7.1

Simplify the numerator.

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Step 2.5.7.1.1

Apply the distributive property.

$y = \frac{- (20 x) + 1040 \pm \sqrt{{(20 x - 1040)}^{2} - 4 \cdot 95 \cdot (x^{2} - 104 x + 2704)}}{2 \cdot 95}$

Step 2.5.7.1.2

Multiply $20$ by $- 1$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{{(20 x - 1040)}^{2} - 4 \cdot 95 \cdot (x^{2} - 104 x + 2704)}}{2 \cdot 95}$

Step 2.5.7.1.3

Multiply $- 1$ by $- 1040$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{{(20 x - 1040)}^{2} - 4 \cdot 95 \cdot (x^{2} - 104 x + 2704)}}{2 \cdot 95}$

Step 2.5.7.1.4

Add parentheses.

$y = \frac{- 20 x + 1040 \pm \sqrt{{(20 x - 1040)}^{2} - 4 \cdot (95 \cdot (x^{2} - 104 x + 2704))}}{2 \cdot 95}$

Step 2.5.7.1.5

Let $u = 95 \cdot (x^{2} - 104 x + 2704)$ . Substitute $u$ for all occurrences of $95 \cdot (x^{2} - 104 x + 2704)$ .

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Step 2.5.7.1.5.1

Rewrite ${(20 x - 1040)}^{2}$ as $(20 x - 1040) (20 x - 1040)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{(20 x - 1040) (20 x - 1040) - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.7.1.5.2

Expand $(20 x - 1040) (20 x - 1040)$ using the FOIL Method.

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Step 2.5.7.1.5.2.1

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{20 x (20 x - 1040) - 1040 (20 x - 1040) - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.7.1.5.2.2

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{20 x (20 x) + 20 x \cdot - 1040 - 1040 (20 x - 1040) - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.7.1.5.2.3

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{20 x (20 x) + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.7.1.5.3

Simplify and combine like terms.

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Step 2.5.7.1.5.3.1

Simplify each term.

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Step 2.5.7.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{- 20 x + 1040 \pm \sqrt{20 \cdot (20 x \cdot x) + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.7.1.5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.5.7.1.5.3.1.2.1

Move $x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{20 \cdot (20 (x \cdot x)) + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.7.1.5.3.1.2.2

Multiply $x$ by $x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{20 \cdot (20 x^{2}) + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.7.1.5.3.1.3

Multiply $20$ by $20$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} + 20 x \cdot - 1040 - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.7.1.5.3.1.4

Multiply $- 1040$ by $20$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 20800 x - 1040 (20 x) - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.7.1.5.3.1.5

Multiply $20$ by $- 1040$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 20800 x - 20800 x - 1040 \cdot - 1040 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.7.1.5.3.1.6

Multiply $- 1040$ by $- 1040$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 20800 x - 20800 x + 1081600 - 4 \cdot u}}{2 \cdot 95}$

Step 2.5.7.1.5.3.2

Subtract $20800 x$ from $- 20800 x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 41600 x + 1081600 - 4 \cdot u}}{2 \cdot 95}$

$y = \frac{- 20 x + 1040 \pm \sqrt{400 x^{2} - 41600 x + 1081600 - 4 u}}{2 \cdot 95}$

Step 2.5.7.1.6

Factor $4$ out of $400 x^{2} - 41600 x + 1081600 - 4 u$ .

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Step 2.5.7.1.6.1

Factor $4$ out of $400 x^{2}$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2}) - 41600 x + 1081600 - 4 u}}{2 \cdot 95}$

Step 2.5.7.1.6.2

Factor $4$ out of $- 41600 x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2}) + 4 (- 10400 x) + 1081600 - 4 u}}{2 \cdot 95}$

Step 2.5.7.1.6.3

Factor $4$ out of $1081600$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2}) + 4 (- 10400 x) + 4 (270400) - 4 u}}{2 \cdot 95}$

Step 2.5.7.1.6.4

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

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2}) + 4 (- 10400 x) + 4 (270400) + 4 (- u)}}{2 \cdot 95}$

Step 2.5.7.1.6.5

Factor $4$ out of $4 (100 x^{2}) + 4 (- 10400 x)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x) + 4 (270400) + 4 (- u)}}{2 \cdot 95}$

Step 2.5.7.1.6.6

Factor $4$ out of $4 (100 x^{2} - 10400 x) + 4 (270400)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400) + 4 (- u)}}{2 \cdot 95}$

Step 2.5.7.1.6.7

Factor $4$ out of $4 (100 x^{2} - 10400 x + 270400) + 4 (- u)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - u)}}{2 \cdot 95}$

Step 2.5.7.1.7

Replace all occurrences of $u$ with $95 \cdot (x^{2} - 104 x + 2704)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 \cdot (x^{2} - 104 x + 2704)))}}{2 \cdot 95}$

Step 2.5.7.1.8

Simplify.

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Step 2.5.7.1.8.1

Simplify each term.

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Step 2.5.7.1.8.1.1

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 x^{2} + 95 (- 104 x) + 95 \cdot 2704))}}{2 \cdot 95}$

Step 2.5.7.1.8.1.2

Simplify.

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Step 2.5.7.1.8.1.2.1

Multiply $- 104$ by $95$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 x^{2} - 9880 x + 95 \cdot 2704))}}{2 \cdot 95}$

Step 2.5.7.1.8.1.2.2

Multiply $95$ by $2704$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 x^{2} - 9880 x + 256880))}}{2 \cdot 95}$

Step 2.5.7.1.8.1.3

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - (95 x^{2}) - (- 9880 x) - 1 \cdot 256880)}}{2 \cdot 95}$

Step 2.5.7.1.8.1.4

Simplify.

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Step 2.5.7.1.8.1.4.1

Multiply $95$ by $- 1$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - 95 x^{2} - (- 9880 x) - 1 \cdot 256880)}}{2 \cdot 95}$

Step 2.5.7.1.8.1.4.2

Multiply $- 9880$ by $- 1$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - 95 x^{2} + 9880 x - 1 \cdot 256880)}}{2 \cdot 95}$

Step 2.5.7.1.8.1.4.3

Multiply $- 1$ by $256880$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (100 x^{2} - 10400 x + 270400 - 95 x^{2} + 9880 x - 256880)}}{2 \cdot 95}$

Step 2.5.7.1.8.2

Subtract $95 x^{2}$ from $100 x^{2}$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 x^{2} - 10400 x + 270400 + 9880 x - 256880)}}{2 \cdot 95}$

Step 2.5.7.1.8.3

Add $- 10400 x$ and $9880 x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 x^{2} - 520 x + 270400 - 256880)}}{2 \cdot 95}$

Step 2.5.7.1.8.4

Subtract $256880$ from $270400$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 x^{2} - 520 x + 13520)}}{2 \cdot 95}$

Step 2.5.7.1.9

Factor $5$ out of $5 x^{2} - 520 x + 13520$ .

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Step 2.5.7.1.9.1

Factor $5$ out of $5 x^{2}$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2}) - 520 x + 13520)}}{2 \cdot 95}$

Step 2.5.7.1.9.2

Factor $5$ out of $- 520 x$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2}) + 5 (- 104 x) + 13520)}}{2 \cdot 95}$

Step 2.5.7.1.9.3

Factor $5$ out of $13520$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 x^{2} + 5 (- 104 x) + 5 \cdot 2704)}}{2 \cdot 95}$

Step 2.5.7.1.9.4

Factor $5$ out of $5 x^{2} + 5 (- 104 x)$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2} - 104 x) + 5 \cdot 2704)}}{2 \cdot 95}$

Step 2.5.7.1.9.5

Factor $5$ out of $5 (x^{2} - 104 x) + 5 \cdot 2704$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2} - 104 x + 2704))}}{2 \cdot 95}$

Step 2.5.7.1.10

Factor using the perfect square rule.

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Step 2.5.7.1.10.1

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

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2} - 104 x + 52^{2}))}}{2 \cdot 95}$

Step 2.5.7.1.10.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$104 x = 2 \cdot x \cdot 52$

Step 2.5.7.1.10.3

Rewrite the polynomial.

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 (x^{2} - 2 \cdot x \cdot 52 + 52^{2}))}}{2 \cdot 95}$

Step 2.5.7.1.10.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 52$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5 {(x - 52)}^{2})}}{2 \cdot 95}$

$y = \frac{- 20 x + 1040 \pm \sqrt{4 \cdot (5 {(x - 52)}^{2})}}{2 \cdot 95}$

Step 2.5.7.1.11

Multiply $4$ by $5$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{20 {(x - 52)}^{2}}}{2 \cdot 95}$

Step 2.5.7.1.12

Rewrite $20 {(x - 52)}^{2}$ as ${(2 (x - 52))}^{2} \cdot 5$ .

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Step 2.5.7.1.12.1

Factor $4$ out of $20$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{4 (5) {(x - 52)}^{2}}}{2 \cdot 95}$

Step 2.5.7.1.12.2

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

$y = \frac{- 20 x + 1040 \pm \sqrt{2^{2} \cdot (5 {(x - 52)}^{2})}}{2 \cdot 95}$

Step 2.5.7.1.12.3

Move $5$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{2^{2} {(x - 52)}^{2} \cdot 5}}{2 \cdot 95}$

Step 2.5.7.1.12.4

Rewrite $2^{2} {(x - 52)}^{2}$ as ${(2 (x - 52))}^{2}$ .

$y = \frac{- 20 x + 1040 \pm \sqrt{{(2 (x - 52))}^{2} \cdot 5}}{2 \cdot 95}$

Step 2.5.7.1.13

Pull terms out from under the radical.

$y = \frac{- 20 x + 1040 \pm 2 (x - 52) \sqrt{5}}{2 \cdot 95}$

Step 2.5.7.1.14

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm (2 x + 2 \cdot - 52) \sqrt{5}}{2 \cdot 95}$

Step 2.5.7.1.15

Multiply $2$ by $- 52$ .

$y = \frac{- 20 x + 1040 \pm (2 x - 104) \sqrt{5}}{2 \cdot 95}$

Step 2.5.7.1.16

Apply the distributive property.

$y = \frac{- 20 x + 1040 \pm (2 x \sqrt{5} - 104 \sqrt{5})}{2 \cdot 95}$

Step 2.5.7.2

Multiply $2$ by $95$ .

$y = \frac{- 20 x + 1040 \pm (2 x \sqrt{5} - 104 \sqrt{5})}{190}$

Step 2.5.7.3

Change the $\pm$ to $-$ .

$y = \frac{- 20 x + 1040 - (2 x \sqrt{5} - 104 \sqrt{5})}{190}$

Step 2.5.7.4

Cancel the common factor of $- 20 x + 1040 - (2 x \sqrt{5} - 104 \sqrt{5})$ and $190$ .

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Step 2.5.7.4.1

Factor $- 1$ out of $- 20 x$ .

$y = \frac{- (20 x) + 1040 - (2 x \sqrt{5} - 104 \sqrt{5})}{190}$

Step 2.5.7.4.2

Rewrite $1040$ as $- 1 (- 1040)$ .

$y = \frac{- (20 x) - 1 \cdot - 1040 - (2 x \sqrt{5} - 104 \sqrt{5})}{190}$

Step 2.5.7.4.3

Factor $- 1$ out of $- (20 x) - 1 (- 1040)$ .

$y = \frac{- (20 x - 1040) - (2 x \sqrt{5} - 104 \sqrt{5})}{190}$

Step 2.5.7.4.4

Factor $- 1$ out of $- (20 x - 1040) - (2 x \sqrt{5} - 104 \sqrt{5})$ .

$y = \frac{- (20 x - 1040 + 2 x \sqrt{5} - 104 \sqrt{5})}{190}$

Step 2.5.7.4.5

Rewrite $- (20 x - 1040 + 2 x \sqrt{5} - 104 \sqrt{5})$ as $- 1 (20 x - 1040 + 2 x \sqrt{5} - 104 \sqrt{5})$ .

$y = \frac{- 1 (20 x - 1040 + 2 x \sqrt{5} - 104 \sqrt{5})}{190}$

Step 2.5.7.4.6

Factor $2$ out of $- 1 (20 x - 1040 + 2 x \sqrt{5} - 104 \sqrt{5})$ .

$y = \frac{2 (- 1 (10 x - 520 + x \sqrt{5} - 52 \sqrt{5}))}{190}$

Step 2.5.7.4.7

Cancel the common factors.

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Step 2.5.7.4.7.1

Factor $2$ out of $190$ .

$y = \frac{2 (- 1 (10 x - 520 + x \sqrt{5} - 52 \sqrt{5}))}{2 (95)}$

Step 2.5.7.4.7.2

Cancel the common factor.

$y = \frac{2 (- 1 (10 x - 520 + x \sqrt{5} - 52 \sqrt{5}))}{2 \cdot 95}$

Step 2.5.7.4.7.3

Rewrite the expression.

$y = \frac{- 1 (10 x - 520 + x \sqrt{5} - 52 \sqrt{5})}{95}$

Step 2.5.7.5

Reorder terms.

$y = \frac{- 1 (x \sqrt{5} + 10 x - 520 - 52 \sqrt{5})}{95}$

Step 2.5.7.6

Move the negative in front of the fraction.

$y = - \frac{x \sqrt{5} + 10 x - 520 - 52 \sqrt{5}}{95}$

Step 2.5.8

The final answer is the combination of both solutions.

$y = \frac{x \sqrt{5} - 10 x + 520 - 52 \sqrt{5}}{95}$

$y = - \frac{x \sqrt{5} + 10 x - 520 - 52 \sqrt{5}}{95}$

$y = \frac{x \sqrt{5} - 10 x + 520 - 52 \sqrt{5}}{95}$

$y = - \frac{x \sqrt{5} + 10 x - 520 - 52 \sqrt{5}}{95}$

$y = \frac{x \sqrt{5} - 10 x + 520 - 52 \sqrt{5}}{95}$

$y = - \frac{x \sqrt{5} + 10 x - 520 - 52 \sqrt{5}}{95}$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{x \sqrt{5} - 10 x + 520 - 52 \sqrt{5}}{95}, - \frac{x \sqrt{5} + 10 x - 520 - 52 \sqrt{5}}{95}$

Step 4

Verify if $f^{- 1} (x) = \frac{x \sqrt{5} - 10 x + 520 - 52 \sqrt{5}}{95}, - \frac{x \sqrt{5} + 10 x - 520 - 52 \sqrt{5}}{95}$ is the inverse of $f (x) = \sqrt{5 x^{2}} - 10 x + 52$ .

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Step 4.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = \sqrt{5 x^{2}} - 10 x + 52$ and $f^{- 1} (x) = \frac{x \sqrt{5} - 10 x + 520 - 52 \sqrt{5}}{95}, - \frac{x \sqrt{5} + 10 x - 520 - 52 \sqrt{5}}{95}$ and compare them.

Step 4.2

Find the range of $f (x) = \sqrt{5 x^{2}} - 10 x + 52$ .

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Step 4.2.1

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Step 4.3

Find the domain of $\frac{x \sqrt{5} - 10 x + 520 - 52 \sqrt{5}}{95}$ .

$(- \infty, \infty)$

Step 4.4

Since the domain of $f^{- 1} (x) = \frac{x \sqrt{5} - 10 x + 520 - 52 \sqrt{5}}{95}, - \frac{x \sqrt{5} + 10 x - 520 - 52 \sqrt{5}}{95}$ is the range of $f (x) = \sqrt{5 x^{2}} - 10 x + 52$ and the range of $f^{- 1} (x) = \frac{x \sqrt{5} - 10 x + 520 - 52 \sqrt{5}}{95}, - \frac{x \sqrt{5} + 10 x - 520 - 52 \sqrt{5}}{95}$ is the domain of $f (x) = \sqrt{5 x^{2}} - 10 x + 52$ , then $f^{- 1} (x) = \frac{x \sqrt{5} - 10 x + 520 - 52 \sqrt{5}}{95}, - \frac{x \sqrt{5} + 10 x - 520 - 52 \sqrt{5}}{95}$ is the inverse of $f (x) = \sqrt{5 x^{2}} - 10 x + 52$ .

$f^{- 1} (x) = \frac{x \sqrt{5} - 10 x + 520 - 52 \sqrt{5}}{95}, - \frac{x \sqrt{5} + 10 x - 520 - 52 \sqrt{5}}{95}$

Step 5