Trigonometry Examples

$y = \frac{x + 7}{24 - \sqrt{x^{2} - 49}}$

Step 1

Rewrite the equation as $\frac{x + 7}{24 - \sqrt{x^{2} - 49}} = y$ .

$\frac{x + 7}{24 - \sqrt{x^{2} - 49}} = y$

Step 2

Multiply both sides by $24 - \sqrt{x^{2} - 49}$ .

$\frac{x + 7}{24 - \sqrt{x^{2} - 49}} (24 - \sqrt{x^{2} - 49}) = y (24 - \sqrt{x^{2} - 49})$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $24 - \sqrt{x^{2} - 49}$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{x + 7}{24 - \sqrt{x^{2} - 49}} (24 - \sqrt{x^{2} - 49}) = y (24 - \sqrt{x^{2} - 49})$

Step 3.1.1.2

Rewrite the expression.

$x + 7 = y (24 - \sqrt{x^{2} - 49})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $y (24 - \sqrt{x^{2} - 49})$ .

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

Rewrite $49$ as $7^{2}$ .

$x + 7 = y (24 - \sqrt{x^{2} - 7^{2}})$

Step 3.2.1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 7$ .

$x + 7 = y (24 - \sqrt{(x + 7) (x - 7)})$

Step 3.2.1.2

Simplify by multiplying through.

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Step 3.2.1.2.1

Apply the distributive property.

$x + 7 = y \cdot 24 + y (- \sqrt{(x + 7) (x - 7)})$

Step 3.2.1.2.2

Simplify the expression.

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Step 3.2.1.2.2.1

Move $24$ to the left of $y$ .

$x + 7 = 24 \cdot y + y (- \sqrt{(x + 7) (x - 7)})$

Step 3.2.1.2.2.2

Rewrite using the commutative property of multiplication.

$x + 7 = 24 \cdot y - y \sqrt{(x + 7) (x - 7)}$

Step 3.2.1.2.2.3

Reorder $24 y$ and $- y \sqrt{(x + 7) (x - 7)}$ .

$x + 7 = - y \sqrt{(x + 7) (x - 7)} + 24 y$

Step 4

Solve for $x$ .

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Step 4.1

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

$- y \sqrt{(x + 7) (x - 7)} + 24 y = x + 7$

Step 4.2

Subtract $24 y$ from both sides of the equation.

$- y \sqrt{(x + 7) (x - 7)} = x + 7 - 24 y$

Step 4.3

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

${(- y \sqrt{(x + 7) (x - 7)})}^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4

Simplify each side of the equation.

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Step 4.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x + 7) (x - 7)}$ as ${((x + 7) (x - 7))}^{\frac{1}{2}}$ .

${(- y {((x + 7) (x - 7))}^{\frac{1}{2}})}^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Simplify ${(- y {((x + 7) (x - 7))}^{\frac{1}{2}})}^{2}$ .

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Step 4.4.2.1.1

Expand $(x + 7) (x - 7)$ using the FOIL Method.

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Step 4.4.2.1.1.1

Apply the distributive property.

${(- y {(x (x - 7) + 7 (x - 7))}^{\frac{1}{2}})}^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.1.2

Apply the distributive property.

${(- y {(x \cdot x + x \cdot - 7 + 7 (x - 7))}^{\frac{1}{2}})}^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.1.3

Apply the distributive property.

${(- y {(x \cdot x + x \cdot - 7 + 7 x + 7 \cdot - 7)}^{\frac{1}{2}})}^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.2

Simplify terms.

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Step 4.4.2.1.2.1

Combine the opposite terms in $x \cdot x + x \cdot - 7 + 7 x + 7 \cdot - 7$ .

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Step 4.4.2.1.2.1.1

Reorder the factors in the terms $x \cdot - 7$ and $7 x$ .

${(- y {(x \cdot x - 7 x + 7 x + 7 \cdot - 7)}^{\frac{1}{2}})}^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.2.1.2

Add $- 7 x$ and $7 x$ .

${(- y {(x \cdot x + 0 + 7 \cdot - 7)}^{\frac{1}{2}})}^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.2.1.3

Add $x \cdot x$ and $0$ .

${(- y {(x \cdot x + 7 \cdot - 7)}^{\frac{1}{2}})}^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.2.2

Simplify each term.

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Step 4.4.2.1.2.2.1

Multiply $x$ by $x$ .

${(- y {(x^{2} + 7 \cdot - 7)}^{\frac{1}{2}})}^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.2.2.2

Multiply $7$ by $- 7$ .

${(- y {(x^{2} - 49)}^{\frac{1}{2}})}^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.4.2.1.3.1

Apply the product rule to $- y {(x^{2} - 49)}^{\frac{1}{2}}$ .

${(- y)}^{2} {({(x^{2} - 49)}^{\frac{1}{2}})}^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.3.2

Apply the product rule to $- y$ .

${(- 1)}^{2} y^{2} {({(x^{2} - 49)}^{\frac{1}{2}})}^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.4

Raise $- 1$ to the power of $2$ .

$1 y^{2} {({(x^{2} - 49)}^{\frac{1}{2}})}^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.5

Multiply $y^{2}$ by $1$ .

$y^{2} {({(x^{2} - 49)}^{\frac{1}{2}})}^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.6

Multiply the exponents in ${({(x^{2} - 49)}^{\frac{1}{2}})}^{2}$ .

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Step 4.4.2.1.6.1

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

$y^{2} {(x^{2} - 49)}^{\frac{1}{2} \cdot 2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.6.2

Cancel the common factor of $2$ .

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Step 4.4.2.1.6.2.1

Cancel the common factor.

$y^{2} {(x^{2} - 49)}^{\frac{1}{2} \cdot 2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.6.2.2

Rewrite the expression.

$y^{2} {(x^{2} - 49)}^{1} = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.7

Simplify.

$y^{2} (x^{2} - 49) = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.8

Apply the distributive property.

$y^{2} x^{2} + y^{2} \cdot - 49 = {(x + 7 - 24 y)}^{2}$

Step 4.4.2.1.9

Move $- 49$ to the left of $y^{2}$ .

$y^{2} x^{2} - 49 y^{2} = {(x + 7 - 24 y)}^{2}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Simplify ${(x + 7 - 24 y)}^{2}$ .

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Step 4.4.3.1.1

Rewrite ${(x + 7 - 24 y)}^{2}$ as $(x + 7 - 24 y) (x + 7 - 24 y)$ .

$y^{2} x^{2} - 49 y^{2} = (x + 7 - 24 y) (x + 7 - 24 y)$

Step 4.4.3.1.2

Expand $(x + 7 - 24 y) (x + 7 - 24 y)$ by multiplying each term in the first expression by each term in the second expression.

$y^{2} x^{2} - 49 y^{2} = x \cdot x + x \cdot 7 + x (- 24 y) + 7 x + 7 \cdot 7 + 7 (- 24 y) - 24 y x - 24 y \cdot 7 - 24 y (- 24 y)$

Step 4.4.3.1.3

Simplify terms.

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Step 4.4.3.1.3.1

Simplify each term.

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Step 4.4.3.1.3.1.1

Multiply $x$ by $x$ .

$y^{2} x^{2} - 49 y^{2} = x^{2} + x \cdot 7 + x (- 24 y) + 7 x + 7 \cdot 7 + 7 (- 24 y) - 24 y x - 24 y \cdot 7 - 24 y (- 24 y)$

Step 4.4.3.1.3.1.2

Move $7$ to the left of $x$ .

$y^{2} x^{2} - 49 y^{2} = x^{2} + 7 \cdot x + x (- 24 y) + 7 x + 7 \cdot 7 + 7 (- 24 y) - 24 y x - 24 y \cdot 7 - 24 y (- 24 y)$

Step 4.4.3.1.3.1.3

Rewrite using the commutative property of multiplication.

$y^{2} x^{2} - 49 y^{2} = x^{2} + 7 x - 24 x y + 7 x + 7 \cdot 7 + 7 (- 24 y) - 24 y x - 24 y \cdot 7 - 24 y (- 24 y)$

Step 4.4.3.1.3.1.4

Multiply $7$ by $7$ .

$y^{2} x^{2} - 49 y^{2} = x^{2} + 7 x - 24 x y + 7 x + 49 + 7 (- 24 y) - 24 y x - 24 y \cdot 7 - 24 y (- 24 y)$

Step 4.4.3.1.3.1.5

Multiply $- 24$ by $7$ .

$y^{2} x^{2} - 49 y^{2} = x^{2} + 7 x - 24 x y + 7 x + 49 - 168 y - 24 y x - 24 y \cdot 7 - 24 y (- 24 y)$

Step 4.4.3.1.3.1.6

Multiply $7$ by $- 24$ .

$y^{2} x^{2} - 49 y^{2} = x^{2} + 7 x - 24 x y + 7 x + 49 - 168 y - 24 y x - 168 y - 24 y (- 24 y)$

Step 4.4.3.1.3.1.7

Rewrite using the commutative property of multiplication.

$y^{2} x^{2} - 49 y^{2} = x^{2} + 7 x - 24 x y + 7 x + 49 - 168 y - 24 y x - 168 y - 24 \cdot - 24 y \cdot y$

Step 4.4.3.1.3.1.8

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

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Step 4.4.3.1.3.1.8.1

Move $y$ .

$y^{2} x^{2} - 49 y^{2} = x^{2} + 7 x - 24 x y + 7 x + 49 - 168 y - 24 y x - 168 y - 24 \cdot - 24 (y \cdot y)$

Step 4.4.3.1.3.1.8.2

Multiply $y$ by $y$ .

$y^{2} x^{2} - 49 y^{2} = x^{2} + 7 x - 24 x y + 7 x + 49 - 168 y - 24 y x - 168 y - 24 \cdot - 24 y^{2}$

Step 4.4.3.1.3.1.9

Multiply $- 24$ by $- 24$ .

$y^{2} x^{2} - 49 y^{2} = x^{2} + 7 x - 24 x y + 7 x + 49 - 168 y - 24 y x - 168 y + 576 y^{2}$

Step 4.4.3.1.3.2

Add $7 x$ and $7 x$ .

$y^{2} x^{2} - 49 y^{2} = x^{2} - 24 x y + 14 x + 49 - 168 y - 24 y x - 168 y + 576 y^{2}$

Step 4.4.3.1.4

Subtract $24 y x$ from $- 24 x y$ .

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Step 4.4.3.1.4.1

Move $y$ .

$y^{2} x^{2} - 49 y^{2} = x^{2} - 24 x y - 24 x y + 14 x + 49 - 168 y - 168 y + 576 y^{2}$

Step 4.4.3.1.4.2

Subtract $24 x y$ from $- 24 x y$ .

$y^{2} x^{2} - 49 y^{2} = x^{2} - 48 x y + 14 x + 49 - 168 y - 168 y + 576 y^{2}$

Step 4.4.3.1.5

Subtract $168 y$ from $- 168 y$ .

$y^{2} x^{2} - 49 y^{2} = x^{2} - 48 x y + 14 x + 49 - 336 y + 576 y^{2}$

Step 4.5

Solve for $x$ .

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Step 4.5.1

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

$x^{2} - 48 x y + 14 x + 49 - 336 y + 576 y^{2} = y^{2} x^{2} - 49 y^{2}$

Step 4.5.2

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

$x^{2} - 48 x y + 14 x + 49 - 336 y + 576 y^{2} - y^{2} x^{2} = - 49 y^{2}$

Step 4.5.3

Move all terms to the left side of the equation and simplify.

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Step 4.5.3.1

Add $49 y^{2}$ to both sides of the equation.

$x^{2} - 48 x y + 14 x + 49 - 336 y + 576 y^{2} - y^{2} x^{2} + 49 y^{2} = 0$

Step 4.5.3.2

Add $576 y^{2}$ and $49 y^{2}$ .

$x^{2} - 48 x y + 14 x + 49 - 336 y + 625 y^{2} - y^{2} x^{2} = 0$

Step 4.5.4

Use the quadratic formula to find the solutions.

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

Step 4.5.5

Substitute the values $a = 1 - y^{2}$ , $b = - 48 y + 14$ , and $c = 49 - 336 y + 625 y^{2}$ into the quadratic formula and solve for $x$ .

$\frac{- (- 48 y + 14) \pm \sqrt{{(- 48 y + 14)}^{2} - 4 \cdot ((1 - y^{2}) \cdot (49 - 336 y + 625 y^{2}))}}{2 (1 - y^{2})}$

Step 4.5.6

Simplify.

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Step 4.5.6.1

Simplify the numerator.

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Step 4.5.6.1.1

Apply the distributive property.

$x = \frac{- (- 48 y) - 1 \cdot 14 \pm \sqrt{{(- 48 y + 14)}^{2} - 4 \cdot (1 - y^{2}) \cdot (49 - 336 y + 625 y^{2})}}{2 (1 - y^{2})}$

Step 4.5.6.1.2

Multiply $- 48$ by $- 1$ .

$x = \frac{48 y - 1 \cdot 14 \pm \sqrt{{(- 48 y + 14)}^{2} - 4 \cdot (1 - y^{2}) \cdot (49 - 336 y + 625 y^{2})}}{2 (1 - y^{2})}$

Step 4.5.6.1.3

Multiply $- 1$ by $14$ .

$x = \frac{48 y - 14 \pm \sqrt{{(- 48 y + 14)}^{2} - 4 \cdot (1 - y^{2}) \cdot (49 - 336 y + 625 y^{2})}}{2 (1 - y^{2})}$

Step 4.5.6.1.4

Add parentheses.

$x = \frac{48 y - 14 \pm \sqrt{{(- 48 y + 14)}^{2} - 4 \cdot ((1 - y^{2}) \cdot (49 - 336 y + 625 y^{2}))}}{2 (1 - y^{2})}$

Step 4.5.6.1.5

Let $u = (1 - y^{2}) \cdot (49 - 336 y + 625 y^{2})$ . Substitute $u$ for all occurrences of $(1 - y^{2}) \cdot (49 - 336 y + 625 y^{2})$ .

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Step 4.5.6.1.5.1

Rewrite ${(- 48 y + 14)}^{2}$ as $(- 48 y + 14) (- 48 y + 14)$ .

$x = \frac{48 y - 14 \pm \sqrt{(- 48 y + 14) (- 48 y + 14) - 4 \cdot u}}{2 (1 - y^{2})}$

Step 4.5.6.1.5.2

Expand $(- 48 y + 14) (- 48 y + 14)$ using the FOIL Method.

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Step 4.5.6.1.5.2.1

Apply the distributive property.

$x = \frac{48 y - 14 \pm \sqrt{- 48 y (- 48 y + 14) + 14 (- 48 y + 14) - 4 \cdot u}}{2 (1 - y^{2})}$

Step 4.5.6.1.5.2.2

Apply the distributive property.

$x = \frac{48 y - 14 \pm \sqrt{- 48 y (- 48 y) - 48 y \cdot 14 + 14 (- 48 y + 14) - 4 \cdot u}}{2 (1 - y^{2})}$

Step 4.5.6.1.5.2.3

Apply the distributive property.

$x = \frac{48 y - 14 \pm \sqrt{- 48 y (- 48 y) - 48 y \cdot 14 + 14 (- 48 y) + 14 \cdot 14 - 4 \cdot u}}{2 (1 - y^{2})}$

Step 4.5.6.1.5.3

Simplify and combine like terms.

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Step 4.5.6.1.5.3.1

Simplify each term.

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Step 4.5.6.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{48 y - 14 \pm \sqrt{- 48 \cdot (- 48 y \cdot y) - 48 y \cdot 14 + 14 (- 48 y) + 14 \cdot 14 - 4 \cdot u}}{2 (1 - y^{2})}$

Step 4.5.6.1.5.3.1.2

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

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Step 4.5.6.1.5.3.1.2.1

Move $y$ .

$x = \frac{48 y - 14 \pm \sqrt{- 48 \cdot (- 48 (y \cdot y)) - 48 y \cdot 14 + 14 (- 48 y) + 14 \cdot 14 - 4 \cdot u}}{2 (1 - y^{2})}$

Step 4.5.6.1.5.3.1.2.2

Multiply $y$ by $y$ .

$x = \frac{48 y - 14 \pm \sqrt{- 48 \cdot (- 48 y^{2}) - 48 y \cdot 14 + 14 (- 48 y) + 14 \cdot 14 - 4 \cdot u}}{2 (1 - y^{2})}$

Step 4.5.6.1.5.3.1.3

Multiply $- 48$ by $- 48$ .

$x = \frac{48 y - 14 \pm \sqrt{2304 y^{2} - 48 y \cdot 14 + 14 (- 48 y) + 14 \cdot 14 - 4 \cdot u}}{2 (1 - y^{2})}$

Step 4.5.6.1.5.3.1.4

Multiply $14$ by $- 48$ .

$x = \frac{48 y - 14 \pm \sqrt{2304 y^{2} - 672 y + 14 (- 48 y) + 14 \cdot 14 - 4 \cdot u}}{2 (1 - y^{2})}$

Step 4.5.6.1.5.3.1.5

Multiply $- 48$ by $14$ .

$x = \frac{48 y - 14 \pm \sqrt{2304 y^{2} - 672 y - 672 y + 14 \cdot 14 - 4 \cdot u}}{2 (1 - y^{2})}$

Step 4.5.6.1.5.3.1.6

Multiply $14$ by $14$ .

$x = \frac{48 y - 14 \pm \sqrt{2304 y^{2} - 672 y - 672 y + 196 - 4 \cdot u}}{2 (1 - y^{2})}$

Step 4.5.6.1.5.3.2

Subtract $672 y$ from $- 672 y$ .

$x = \frac{48 y - 14 \pm \sqrt{2304 y^{2} - 1344 y + 196 - 4 \cdot u}}{2 (1 - y^{2})}$

$x = \frac{48 y - 14 \pm \sqrt{2304 y^{2} - 1344 y + 196 - 4 u}}{2 (1 - y^{2})}$

Step 4.5.6.1.6

Factor $4$ out of $2304 y^{2} - 1344 y + 196 - 4 u$ .

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Step 4.5.6.1.6.1

Factor $4$ out of $2304 y^{2}$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2}) - 1344 y + 196 - 4 u}}{2 (1 - y^{2})}$

Step 4.5.6.1.6.2

Factor $4$ out of $- 1344 y$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2}) + 4 (- 336 y) + 196 - 4 u}}{2 (1 - y^{2})}$

Step 4.5.6.1.6.3

Factor $4$ out of $196$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2}) + 4 (- 336 y) + 4 (49) - 4 u}}{2 (1 - y^{2})}$

Step 4.5.6.1.6.4

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

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2}) + 4 (- 336 y) + 4 (49) + 4 (- u)}}{2 (1 - y^{2})}$

Step 4.5.6.1.6.5

Factor $4$ out of $4 (576 y^{2}) + 4 (- 336 y)$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y) + 4 (49) + 4 (- u)}}{2 (1 - y^{2})}$

Step 4.5.6.1.6.6

Factor $4$ out of $4 (576 y^{2} - 336 y) + 4 (49)$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49) + 4 (- u)}}{2 (1 - y^{2})}$

Step 4.5.6.1.6.7

Factor $4$ out of $4 (576 y^{2} - 336 y + 49) + 4 (- u)$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - u)}}{2 (1 - y^{2})}$

Step 4.5.6.1.7

Replace all occurrences of $u$ with $(1 - y^{2}) \cdot (49 - 336 y + 625 y^{2})$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - ((1 - y^{2}) \cdot (49 - 336 y + 625 y^{2})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8

Simplify.

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Step 4.5.6.1.8.1

Simplify each term.

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Step 4.5.6.1.8.1.1

Expand $(1 - y^{2}) (49 - 336 y + 625 y^{2})$ by multiplying each term in the first expression by each term in the second expression.

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (1 \cdot 49 + 1 (- 336 y) + 1 (625 y^{2}) - y^{2} \cdot 49 - y^{2} (- 336 y) - y^{2} (625 y^{2})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2

Simplify each term.

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Step 4.5.6.1.8.1.2.1

Multiply $49$ by $1$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 + 1 (- 336 y) + 1 (625 y^{2}) - y^{2} \cdot 49 - y^{2} (- 336 y) - y^{2} (625 y^{2})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2.2

Multiply $- 336 y$ by $1$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 1 (625 y^{2}) - y^{2} \cdot 49 - y^{2} (- 336 y) - y^{2} (625 y^{2})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2.3

Multiply $625 y^{2}$ by $1$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 625 y^{2} - y^{2} \cdot 49 - y^{2} (- 336 y) - y^{2} (625 y^{2})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2.4

Multiply $49$ by $- 1$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 625 y^{2} - 49 y^{2} - y^{2} (- 336 y) - y^{2} (625 y^{2})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2.5

Rewrite using the commutative property of multiplication.

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 625 y^{2} - 49 y^{2} - 1 \cdot (- 336 y^{2} y) - y^{2} (625 y^{2})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2.6

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 4.5.6.1.8.1.2.6.1

Move $y$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 625 y^{2} - 49 y^{2} - 1 \cdot (- 336 (y \cdot y^{2})) - y^{2} (625 y^{2})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2.6.2

Multiply $y$ by $y^{2}$ .

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Step 4.5.6.1.8.1.2.6.2.1

Raise $y$ to the power of $1$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 625 y^{2} - 49 y^{2} - 1 \cdot (- 336 (y \cdot y^{2})) - y^{2} (625 y^{2})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 625 y^{2} - 49 y^{2} - 1 \cdot (- 336 y^{1 + 2}) - y^{2} (625 y^{2})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2.6.3

Add $1$ and $2$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 625 y^{2} - 49 y^{2} - 1 \cdot (- 336 y^{3}) - y^{2} (625 y^{2})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2.7

Multiply $- 1$ by $- 336$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 625 y^{2} - 49 y^{2} + 336 y^{3} - y^{2} (625 y^{2})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2.8

Rewrite using the commutative property of multiplication.

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 625 y^{2} - 49 y^{2} + 336 y^{3} - 1 \cdot (625 y^{2} y^{2})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2.9

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 4.5.6.1.8.1.2.9.1

Move $y^{2}$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 625 y^{2} - 49 y^{2} + 336 y^{3} - 1 \cdot (625 (y^{2} y^{2}))))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2.9.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 625 y^{2} - 49 y^{2} + 336 y^{3} - 1 \cdot (625 y^{2 + 2})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2.9.3

Add $2$ and $2$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 625 y^{2} - 49 y^{2} + 336 y^{3} - 1 \cdot (625 y^{4})))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.2.10

Multiply $- 1$ by $625$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 625 y^{2} - 49 y^{2} + 336 y^{3} - 625 y^{4}))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.3

Subtract $49 y^{2}$ from $625 y^{2}$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - (49 - 336 y + 576 y^{2} + 336 y^{3} - 625 y^{4}))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.4

Apply the distributive property.

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - 1 \cdot 49 - (- 336 y) - (576 y^{2}) - (336 y^{3}) - (- 625 y^{4}))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.5

Simplify.

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Step 4.5.6.1.8.1.5.1

Multiply $- 1$ by $49$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - 49 - (- 336 y) - (576 y^{2}) - (336 y^{3}) - (- 625 y^{4}))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.5.2

Multiply $- 336$ by $- 1$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - 49 + 336 y - (576 y^{2}) - (336 y^{3}) - (- 625 y^{4}))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.5.3

Multiply $576$ by $- 1$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - 49 + 336 y - 576 y^{2} - (336 y^{3}) - (- 625 y^{4}))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.5.4

Multiply $336$ by $- 1$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - 49 + 336 y - 576 y^{2} - 336 y^{3} - (- 625 y^{4}))}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.1.5.5

Multiply $- 625$ by $- 1$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 49 - 49 + 336 y - 576 y^{2} - 336 y^{3} + 625 y^{4})}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.2

Combine the opposite terms in $576 y^{2} - 336 y + 49 - 49 + 336 y - 576 y^{2} - 336 y^{3} + 625 y^{4}$ .

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Step 4.5.6.1.8.2.1

Subtract $49$ from $49$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 0 + 336 y - 576 y^{2} - 336 y^{3} + 625 y^{4})}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.2.2

Add $576 y^{2} - 336 y$ and $0$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 336 y + 336 y - 576 y^{2} - 336 y^{3} + 625 y^{4})}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.2.3

Add $- 336 y$ and $336 y$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} + 0 - 576 y^{2} - 336 y^{3} + 625 y^{4})}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.2.4

Add $576 y^{2}$ and $0$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (576 y^{2} - 576 y^{2} - 336 y^{3} + 625 y^{4})}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.2.5

Subtract $576 y^{2}$ from $576 y^{2}$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (0 - 336 y^{3} + 625 y^{4})}}{2 (1 - y^{2})}$

Step 4.5.6.1.8.3

Subtract $336 y^{3}$ from $0$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (- 336 y^{3} + 625 y^{4})}}{2 (1 - y^{2})}$

Step 4.5.6.1.9

Factor $y^{3}$ out of $- 336 y^{3} + 625 y^{4}$ .

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Step 4.5.6.1.9.1

Factor $y^{3}$ out of $- 336 y^{3}$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (y^{3} \cdot - 336 + 625 y^{4})}}{2 (1 - y^{2})}$

Step 4.5.6.1.9.2

Factor $y^{3}$ out of $625 y^{4}$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (y^{3} \cdot - 336 + y^{3} (625 y))}}{2 (1 - y^{2})}$

Step 4.5.6.1.9.3

Factor $y^{3}$ out of $y^{3} \cdot - 336 + y^{3} (625 y)$ .

$x = \frac{48 y - 14 \pm \sqrt{4 (y^{3} (- 336 + 625 y))}}{2 (1 - y^{2})}$

$x = \frac{48 y - 14 \pm \sqrt{4 y^{3} (- 336 + 625 y)}}{2 (1 - y^{2})}$

Step 4.5.6.1.10

Rewrite $4 y^{3} (- 336 + 625 y)$ as ${(2 y)}^{2} (y (- 336 + 625 y))$ .

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Step 4.5.6.1.10.1

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

$x = \frac{48 y - 14 \pm \sqrt{2^{2} y^{3} (- 336 + 625 y)}}{2 (1 - y^{2})}$

Step 4.5.6.1.10.2

Factor out $y^{2}$ .

$x = \frac{48 y - 14 \pm \sqrt{2^{2} (y^{2} y) (- 336 + 625 y)}}{2 (1 - y^{2})}$

Step 4.5.6.1.10.3

Rewrite $2^{2} y^{2}$ as ${(2 y)}^{2}$ .

$x = \frac{48 y - 14 \pm \sqrt{{(2 y)}^{2} y (- 336 + 625 y)}}{2 (1 - y^{2})}$

Step 4.5.6.1.10.4

Add parentheses.

$x = \frac{48 y - 14 \pm \sqrt{{(2 y)}^{2} (y (- 336 + 625 y))}}{2 (1 - y^{2})}$

Step 4.5.6.1.11

Pull terms out from under the radical.

$x = \frac{48 y - 14 \pm 2 y \sqrt{y (- 336 + 625 y)}}{2 (1 - y^{2})}$

Step 4.5.6.2

Simplify the denominator.

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Step 4.5.6.2.1

Rewrite $1$ as $1^{2}$ .

$x = \frac{48 y - 14 \pm 2 y \sqrt{y (- 336 + 625 y)}}{2 (1^{2} - y^{2})}$

Step 4.5.6.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = y$ .

$x = \frac{48 y - 14 \pm 2 y \sqrt{y (- 336 + 625 y)}}{2 (1 + y) (1 - y)}$

Step 4.5.7

The final answer is the combination of both solutions.

$x = \frac{24 y - 7 + y \sqrt{y (- 336 + 625 y)}}{(1 + y) (1 - y)}$

$x = \frac{24 y - 7 - y \sqrt{y (- 336 + 625 y)}}{(1 + y) (1 - y)}$

$x = \frac{24 y - 7 + y \sqrt{y (- 336 + 625 y)}}{(1 + y) (1 - y)}$

$x = \frac{24 y - 7 - y \sqrt{y (- 336 + 625 y)}}{(1 + y) (1 - y)}$

$x = \frac{24 y - 7 + y \sqrt{y (- 336 + 625 y)}}{(1 + y) (1 - y)}$

$x = \frac{24 y - 7 - y \sqrt{y (- 336 + 625 y)}}{(1 + y) (1 - y)}$