Finite Math Examples

$\sqrt{4 y + 20} - v y - 4 = 6$

Step 1

Move all terms not containing $\sqrt{4 y + 20}$ to the right side of the equation.

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Step 1.1

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

$\sqrt{4 y + 20} - 4 = 6 + v y$

Step 1.2

Add $4$ to both sides of the equation.

$\sqrt{4 y + 20} = 6 + v y + 4$

Step 1.3

Add $6$ and $4$ .

$\sqrt{4 y + 20} = v y + 10$

Step 2

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

${\sqrt{4 y + 20}}^{2} = {(v y + 10)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 y + 20}$ as ${(4 y + 20)}^{\frac{1}{2}}$ .

${({(4 y + 20)}^{\frac{1}{2}})}^{2} = {(v y + 10)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(4 y + 20)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(4 y + 20)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(4 y + 20)}^{\frac{1}{2} \cdot 2} = {(v y + 10)}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(4 y + 20)}^{\frac{1}{2} \cdot 2} = {(v y + 10)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(4 y + 20)}^{1} = {(v y + 10)}^{2}$

Step 3.2.1.2

Simplify.

$4 y + 20 = {(v y + 10)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(v y + 10)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(v y + 10)}^{2}$ as $(v y + 10) (v y + 10)$ .

$4 y + 20 = (v y + 10) (v y + 10)$

Step 3.3.1.2

Expand $(v y + 10) (v y + 10)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$4 y + 20 = v y (v y + 10) + 10 (v y + 10)$

Step 3.3.1.2.2

Apply the distributive property.

$4 y + 20 = v y (v y) + v y \cdot 10 + 10 (v y + 10)$

Step 3.3.1.2.3

Apply the distributive property.

$4 y + 20 = v y (v y) + v y \cdot 10 + 10 (v y) + 10 \cdot 10$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $v$ by $v$ by adding the exponents.

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Step 3.3.1.3.1.1.1

Move $v$ .

$4 y + 20 = v \cdot v y \cdot y + v y \cdot 10 + 10 (v y) + 10 \cdot 10$

Step 3.3.1.3.1.1.2

Multiply $v$ by $v$ .

$4 y + 20 = v^{2} y \cdot y + v y \cdot 10 + 10 (v y) + 10 \cdot 10$

Step 3.3.1.3.1.2

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

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Step 3.3.1.3.1.2.1

Move $y$ .

$4 y + 20 = v^{2} (y \cdot y) + v y \cdot 10 + 10 (v y) + 10 \cdot 10$

Step 3.3.1.3.1.2.2

Multiply $y$ by $y$ .

$4 y + 20 = v^{2} y^{2} + v y \cdot 10 + 10 (v y) + 10 \cdot 10$

Step 3.3.1.3.1.3

Move $10$ to the left of $v y$ .

$4 y + 20 = v^{2} y^{2} + 10 \cdot (v y) + 10 (v y) + 10 \cdot 10$

Step 3.3.1.3.1.4

Multiply $10$ by $10$ .

$4 y + 20 = v^{2} y^{2} + 10 v y + 10 v y + 100$

Step 3.3.1.3.2

Add $10 v y$ and $10 v y$ .

$4 y + 20 = v^{2} y^{2} + 20 v y + 100$

Step 4

Solve for $y$ .

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Step 4.1

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

$v^{2} y^{2} + 20 v y + 100 = 4 y + 20$

Step 4.2

Subtract $4 y$ from both sides of the equation.

$v^{2} y^{2} + 20 v y + 100 - 4 y = 20$

Step 4.3

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

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Step 4.3.1

Subtract $20$ from both sides of the equation.

$v^{2} y^{2} + 20 v y + 100 - 4 y - 20 = 0$

Step 4.3.2

Subtract $20$ from $100$ .

$v^{2} y^{2} + 20 v y - 4 y + 80 = 0$

Step 4.4

Use the quadratic formula to find the solutions.

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

Step 4.5

Substitute the values $a = v^{2}$ , $b = 20 v - 4$ , and $c = 80$ into the quadratic formula and solve for $y$ .

$\frac{- (20 v - 4) \pm \sqrt{{(20 v - 4)}^{2} - 4 \cdot (v^{2} \cdot 80)}}{2 v^{2}}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Apply the distributive property.

$y = \frac{- (20 v) + 4 \pm \sqrt{{(20 v - 4)}^{2} - 4 \cdot v^{2} \cdot 80}}{2 v^{2}}$

Step 4.6.2

Multiply $20$ by $- 1$ .

$y = \frac{- 20 v + 4 \pm \sqrt{{(20 v - 4)}^{2} - 4 \cdot v^{2} \cdot 80}}{2 v^{2}}$

Step 4.6.3

Multiply $- 1$ by $- 4$ .

$y = \frac{- 20 v + 4 \pm \sqrt{{(20 v - 4)}^{2} - 4 \cdot v^{2} \cdot 80}}{2 v^{2}}$

Step 4.6.4

Add parentheses.

$y = \frac{- 20 v + 4 \pm \sqrt{{(20 v - 4)}^{2} - 4 \cdot (v^{2} \cdot 80)}}{2 v^{2}}$

Step 4.6.5

Let $u = v^{2} \cdot 80$ . Substitute $u$ for all occurrences of $v^{2} \cdot 80$ .

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Step 4.6.5.1

Rewrite ${(20 v - 4)}^{2}$ as $(20 v - 4) (20 v - 4)$ .

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

Step 4.6.5.2

Expand $(20 v - 4) (20 v - 4)$ using the FOIL Method.

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Step 4.6.5.2.1

Apply the distributive property.

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

Step 4.6.5.2.2

Apply the distributive property.

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

Step 4.6.5.2.3

Apply the distributive property.

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

Step 4.6.5.3

Simplify and combine like terms.

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Step 4.6.5.3.1

Simplify each term.

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Step 4.6.5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.6.5.3.1.2

Multiply $v$ by $v$ by adding the exponents.

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Step 4.6.5.3.1.2.1

Move $v$ .

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

Step 4.6.5.3.1.2.2

Multiply $v$ by $v$ .

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

Step 4.6.5.3.1.3

Multiply $20$ by $20$ .

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

Step 4.6.5.3.1.4

Multiply $- 4$ by $20$ .

$y = \frac{- 20 v + 4 \pm \sqrt{400 v^{2} - 80 v - 4 (20 v) - 4 \cdot - 4 - 4 \cdot u}}{2 v^{2}}$

Step 4.6.5.3.1.5

Multiply $20$ by $- 4$ .

$y = \frac{- 20 v + 4 \pm \sqrt{400 v^{2} - 80 v - 80 v - 4 \cdot - 4 - 4 \cdot u}}{2 v^{2}}$

Step 4.6.5.3.1.6

Multiply $- 4$ by $- 4$ .

$y = \frac{- 20 v + 4 \pm \sqrt{400 v^{2} - 80 v - 80 v + 16 - 4 \cdot u}}{2 v^{2}}$

Step 4.6.5.3.2

Subtract $80 v$ from $- 80 v$ .

$y = \frac{- 20 v + 4 \pm \sqrt{400 v^{2} - 160 v + 16 - 4 \cdot u}}{2 v^{2}}$

$y = \frac{- 20 v + 4 \pm \sqrt{400 v^{2} - 160 v + 16 - 4 u}}{2 v^{2}}$

Step 4.6.6

Factor $4$ out of $400 v^{2} - 160 v + 16 - 4 u$ .

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Step 4.6.6.1

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

$y = \frac{- 20 v + 4 \pm \sqrt{4 (100 v^{2}) - 160 v + 16 - 4 u}}{2 v^{2}}$

Step 4.6.6.2

Factor $4$ out of $- 160 v$ .

$y = \frac{- 20 v + 4 \pm \sqrt{4 (100 v^{2}) + 4 (- 40 v) + 16 - 4 u}}{2 v^{2}}$

Step 4.6.6.3

Factor $4$ out of $16$ .

$y = \frac{- 20 v + 4 \pm \sqrt{4 (100 v^{2}) + 4 (- 40 v) + 4 (4) - 4 u}}{2 v^{2}}$

Step 4.6.6.4

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

$y = \frac{- 20 v + 4 \pm \sqrt{4 (100 v^{2}) + 4 (- 40 v) + 4 (4) + 4 (- u)}}{2 v^{2}}$

Step 4.6.6.5

Factor $4$ out of $4 (100 v^{2}) + 4 (- 40 v)$ .

$y = \frac{- 20 v + 4 \pm \sqrt{4 (100 v^{2} - 40 v) + 4 (4) + 4 (- u)}}{2 v^{2}}$

Step 4.6.6.6

Factor $4$ out of $4 (100 v^{2} - 40 v) + 4 (4)$ .

$y = \frac{- 20 v + 4 \pm \sqrt{4 (100 v^{2} - 40 v + 4) + 4 (- u)}}{2 v^{2}}$

Step 4.6.6.7

Factor $4$ out of $4 (100 v^{2} - 40 v + 4) + 4 (- u)$ .

$y = \frac{- 20 v + 4 \pm \sqrt{4 (100 v^{2} - 40 v + 4 - u)}}{2 v^{2}}$

Step 4.6.7

Replace all occurrences of $u$ with $v^{2} \cdot 80$ .

$y = \frac{- 20 v + 4 \pm \sqrt{4 (100 v^{2} - 40 v + 4 - (v^{2} \cdot 80))}}{2 v^{2}}$

Step 4.6.8

Simplify.

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Step 4.6.8.1

Simplify each term.

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Step 4.6.8.1.1

Move $80$ to the left of $v^{2}$ .

$y = \frac{- 20 v + 4 \pm \sqrt{4 (100 v^{2} - 40 v + 4 - (80 \cdot v^{2}))}}{2 v^{2}}$

Step 4.6.8.1.2

Multiply $80$ by $- 1$ .

$y = \frac{- 20 v + 4 \pm \sqrt{4 (100 v^{2} - 40 v + 4 - 80 v^{2})}}{2 v^{2}}$

Step 4.6.8.2

Subtract $80 v^{2}$ from $100 v^{2}$ .

$y = \frac{- 20 v + 4 \pm \sqrt{4 (20 v^{2} - 40 v + 4)}}{2 v^{2}}$

Step 4.6.9

Factor $4$ out of $20 v^{2} - 40 v + 4$ .

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Step 4.6.9.1

Factor $4$ out of $20 v^{2}$ .

$y = \frac{- 20 v + 4 \pm \sqrt{4 (4 (5 v^{2}) - 40 v + 4)}}{2 v^{2}}$

Step 4.6.9.2

Factor $4$ out of $- 40 v$ .

$y = \frac{- 20 v + 4 \pm \sqrt{4 (4 (5 v^{2}) + 4 (- 10 v) + 4)}}{2 v^{2}}$

Step 4.6.9.3

Factor $4$ out of $4$ .

$y = \frac{- 20 v + 4 \pm \sqrt{4 (4 (5 v^{2}) + 4 (- 10 v) + 4 (1))}}{2 v^{2}}$

Step 4.6.9.4

Factor $4$ out of $4 (5 v^{2}) + 4 (- 10 v)$ .

$y = \frac{- 20 v + 4 \pm \sqrt{4 (4 (5 v^{2} - 10 v) + 4 (1))}}{2 v^{2}}$

Step 4.6.9.5

Factor $4$ out of $4 (5 v^{2} - 10 v) + 4 (1)$ .

$y = \frac{- 20 v + 4 \pm \sqrt{4 (4 (5 v^{2} - 10 v + 1))}}{2 v^{2}}$

$y = \frac{- 20 v + 4 \pm \sqrt{4 \cdot (4 (5 v^{2} - 10 v + 1))}}{2 v^{2}}$

Step 4.6.10

Multiply $4$ by $4$ .

$y = \frac{- 20 v + 4 \pm \sqrt{16 (5 v^{2} - 10 v + 1)}}{2 v^{2}}$

Step 4.6.11

Rewrite $16 (5 v^{2} - 10 v + 1)$ as ${(2^{2})}^{2} (5 v^{2} - 10 v + 1)$ .

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Step 4.6.11.1

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

$y = \frac{- 20 v + 4 \pm \sqrt{4^{2} (5 v^{2} - 10 v + 1)}}{2 v^{2}}$

Step 4.6.11.2

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

$y = \frac{- 20 v + 4 \pm \sqrt{{(2^{2})}^{2} (5 v^{2} - 10 v + 1)}}{2 v^{2}}$

Step 4.6.12

Pull terms out from under the radical.

$y = \frac{- 20 v + 4 \pm 2^{2} \sqrt{5 v^{2} - 10 v + 1}}{2 v^{2}}$

Step 4.6.13

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

$y = \frac{- 20 v + 4 \pm 4 \sqrt{5 v^{2} - 10 v + 1}}{2 v^{2}}$

Step 4.7

The final answer is the combination of both solutions.

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

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

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

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