Algebra Examples

$\sqrt{1 - x} - 6 y = 5$

Step 1

Add $6 y$ to both sides of the equation.

$\sqrt{1 - x} = 5 + 6 y$

Step 2

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

${\sqrt{1 - x}}^{2} = {(5 + 6 y)}^{2}$

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 $\sqrt{1 - x}$ as ${(1 - x)}^{\frac{1}{2}}$ .

${({(1 - x)}^{\frac{1}{2}})}^{2} = {(5 + 6 y)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

${(1 - x)}^{\frac{1}{2} \cdot 2} = {(5 + 6 y)}^{2}$

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.

${(1 - x)}^{\frac{1}{2} \cdot 2} = {(5 + 6 y)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(1 - x)}^{1} = {(5 + 6 y)}^{2}$

Step 3.2.1.2

Simplify.

$1 - x = {(5 + 6 y)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(5 + 6 y)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(5 + 6 y)}^{2}$ as $(5 + 6 y) (5 + 6 y)$ .

$1 - x = (5 + 6 y) (5 + 6 y)$

Step 3.3.1.2

Expand $(5 + 6 y) (5 + 6 y)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$1 - x = 5 (5 + 6 y) + 6 y (5 + 6 y)$

Step 3.3.1.2.2

Apply the distributive property.

$1 - x = 5 \cdot 5 + 5 (6 y) + 6 y (5 + 6 y)$

Step 3.3.1.2.3

Apply the distributive property.

$1 - x = 5 \cdot 5 + 5 (6 y) + 6 y \cdot 5 + 6 y (6 y)$

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 $5$ by $5$ .

$1 - x = 25 + 5 (6 y) + 6 y \cdot 5 + 6 y (6 y)$

Step 3.3.1.3.1.2

Multiply $6$ by $5$ .

$1 - x = 25 + 30 y + 6 y \cdot 5 + 6 y (6 y)$

Step 3.3.1.3.1.3

Multiply $5$ by $6$ .

$1 - x = 25 + 30 y + 30 y + 6 y (6 y)$

Step 3.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$1 - x = 25 + 30 y + 30 y + 6 \cdot 6 y \cdot y$

Step 3.3.1.3.1.5

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

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Step 3.3.1.3.1.5.1

Move $y$ .

$1 - x = 25 + 30 y + 30 y + 6 \cdot 6 (y \cdot y)$

Step 3.3.1.3.1.5.2

Multiply $y$ by $y$ .

$1 - x = 25 + 30 y + 30 y + 6 \cdot 6 y^{2}$

Step 3.3.1.3.1.6

Multiply $6$ by $6$ .

$1 - x = 25 + 30 y + 30 y + 36 y^{2}$

Step 3.3.1.3.2

Add $30 y$ and $30 y$ .

$1 - x = 25 + 60 y + 36 y^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1.1

Subtract $1$ from both sides of the equation.

$- x = 25 + 60 y + 36 y^{2} - 1$

Step 4.1.2

Subtract $1$ from $25$ .

$- x = 60 y + 36 y^{2} + 24$

Step 4.2

Divide each term in $- x = 60 y + 36 y^{2} + 24$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- x = 60 y + 36 y^{2} + 24$ by $- 1$ .

$\frac{- x}{- 1} = \frac{60 y}{- 1} + \frac{36 y^{2}}{- 1} + \frac{24}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{60 y}{- 1} + \frac{36 y^{2}}{- 1} + \frac{24}{- 1}$

Step 4.2.2.2

Divide $x$ by $1$ .

$x = \frac{60 y}{- 1} + \frac{36 y^{2}}{- 1} + \frac{24}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Move the negative one from the denominator of $\frac{60 y}{- 1}$ .

$x = - 1 \cdot (60 y) + \frac{36 y^{2}}{- 1} + \frac{24}{- 1}$

Step 4.2.3.1.2

Rewrite $- 1 \cdot (60 y)$ as $- (60 y)$ .

$x = - (60 y) + \frac{36 y^{2}}{- 1} + \frac{24}{- 1}$

Step 4.2.3.1.3

Multiply $60$ by $- 1$ .

$x = - 60 y + \frac{36 y^{2}}{- 1} + \frac{24}{- 1}$

Step 4.2.3.1.4

Move the negative one from the denominator of $\frac{36 y^{2}}{- 1}$ .

$x = - 60 y - 1 \cdot (36 y^{2}) + \frac{24}{- 1}$

Step 4.2.3.1.5

Rewrite $- 1 \cdot (36 y^{2})$ as $- (36 y^{2})$ .

$x = - 60 y - (36 y^{2}) + \frac{24}{- 1}$

Step 4.2.3.1.6

Multiply $36$ by $- 1$ .

$x = - 60 y - 36 y^{2} + \frac{24}{- 1}$

Step 4.2.3.1.7

Divide $24$ by $- 1$ .

$x = - 60 y - 36 y^{2} - 24$

Step 5

To rewrite as a function of $y$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $y$ is on the other side.

$f (y) = - 60 y - 36 y^{2} - 24$