Precalculus Examples

$4 = \sqrt{- 6 - 2 x} + \sqrt{31 - 3 x}$

Step 1

Rewrite the equation as $\sqrt{- 6 - 2 x} + \sqrt{31 - 3 x} = 4$ .

$\sqrt{- 6 - 2 x} + \sqrt{31 - 3 x} = 4$

Step 2

Subtract $\sqrt{31 - 3 x}$ from both sides of the equation.

$\sqrt{- 6 - 2 x} = 4 - \sqrt{31 - 3 x}$

Step 3

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

${\sqrt{- 6 - 2 x}}^{2} = {(4 - \sqrt{31 - 3 x})}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 6 - 2 x}$ as ${(- 6 - 2 x)}^{\frac{1}{2}}$ .

${({(- 6 - 2 x)}^{\frac{1}{2}})}^{2} = {(4 - \sqrt{31 - 3 x})}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${({(- 6 - 2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Multiply the exponents in ${({(- 6 - 2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1.1

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

${(- 6 - 2 x)}^{\frac{1}{2} \cdot 2} = {(4 - \sqrt{31 - 3 x})}^{2}$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

${(- 6 - 2 x)}^{\frac{1}{2} \cdot 2} = {(4 - \sqrt{31 - 3 x})}^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(- 6 - 2 x)}^{1} = {(4 - \sqrt{31 - 3 x})}^{2}$

Step 4.2.1.2

Simplify.

$- 6 - 2 x = {(4 - \sqrt{31 - 3 x})}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(4 - \sqrt{31 - 3 x})}^{2}$ .

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Step 4.3.1.1

Rewrite ${(4 - \sqrt{31 - 3 x})}^{2}$ as $(4 - \sqrt{31 - 3 x}) (4 - \sqrt{31 - 3 x})$ .

$- 6 - 2 x = (4 - \sqrt{31 - 3 x}) (4 - \sqrt{31 - 3 x})$

Step 4.3.1.2

Expand $(4 - \sqrt{31 - 3 x}) (4 - \sqrt{31 - 3 x})$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$- 6 - 2 x = 4 (4 - \sqrt{31 - 3 x}) - \sqrt{31 - 3 x} (4 - \sqrt{31 - 3 x})$

Step 4.3.1.2.2

Apply the distributive property.

$- 6 - 2 x = 4 \cdot 4 + 4 (- \sqrt{31 - 3 x}) - \sqrt{31 - 3 x} (4 - \sqrt{31 - 3 x})$

Step 4.3.1.2.3

Apply the distributive property.

$- 6 - 2 x = 4 \cdot 4 + 4 (- \sqrt{31 - 3 x}) - \sqrt{31 - 3 x} \cdot 4 - \sqrt{31 - 3 x} (- \sqrt{31 - 3 x})$

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Multiply $4$ by $4$ .

$- 6 - 2 x = 16 + 4 (- \sqrt{31 - 3 x}) - \sqrt{31 - 3 x} \cdot 4 - \sqrt{31 - 3 x} (- \sqrt{31 - 3 x})$

Step 4.3.1.3.1.2

Multiply $- 1$ by $4$ .

$- 6 - 2 x = 16 - 4 \sqrt{31 - 3 x} - \sqrt{31 - 3 x} \cdot 4 - \sqrt{31 - 3 x} (- \sqrt{31 - 3 x})$

Step 4.3.1.3.1.3

Multiply $4$ by $- 1$ .

$- 6 - 2 x = 16 - 4 \sqrt{31 - 3 x} - 4 \sqrt{31 - 3 x} - \sqrt{31 - 3 x} (- \sqrt{31 - 3 x})$

Step 4.3.1.3.1.4

Multiply $- \sqrt{31 - 3 x} (- \sqrt{31 - 3 x})$ .

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Step 4.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$- 6 - 2 x = 16 - 4 \sqrt{31 - 3 x} - 4 \sqrt{31 - 3 x} + 1 \sqrt{31 - 3 x} \sqrt{31 - 3 x}$

Step 4.3.1.3.1.4.2

Multiply $\sqrt{31 - 3 x}$ by $1$ .

$- 6 - 2 x = 16 - 4 \sqrt{31 - 3 x} - 4 \sqrt{31 - 3 x} + \sqrt{31 - 3 x} \sqrt{31 - 3 x}$

Step 4.3.1.3.1.4.3

Raise $\sqrt{31 - 3 x}$ to the power of $1$ .

$- 6 - 2 x = 16 - 4 \sqrt{31 - 3 x} - 4 \sqrt{31 - 3 x} + {\sqrt{31 - 3 x}}^{1} \sqrt{31 - 3 x}$

Step 4.3.1.3.1.4.4

Raise $\sqrt{31 - 3 x}$ to the power of $1$ .

$- 6 - 2 x = 16 - 4 \sqrt{31 - 3 x} - 4 \sqrt{31 - 3 x} + {\sqrt{31 - 3 x}}^{1} {\sqrt{31 - 3 x}}^{1}$

Step 4.3.1.3.1.4.5

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

$- 6 - 2 x = 16 - 4 \sqrt{31 - 3 x} - 4 \sqrt{31 - 3 x} + {\sqrt{31 - 3 x}}^{1 + 1}$

Step 4.3.1.3.1.4.6

Add $1$ and $1$ .

$- 6 - 2 x = 16 - 4 \sqrt{31 - 3 x} - 4 \sqrt{31 - 3 x} + {\sqrt{31 - 3 x}}^{2}$

Step 4.3.1.3.1.5

Rewrite ${\sqrt{31 - 3 x}}^{2}$ as $31 - 3 x$ .

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Step 4.3.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{31 - 3 x}$ as ${(31 - 3 x)}^{\frac{1}{2}}$ .

$- 6 - 2 x = 16 - 4 \sqrt{31 - 3 x} - 4 \sqrt{31 - 3 x} + {({(31 - 3 x)}^{\frac{1}{2}})}^{2}$

Step 4.3.1.3.1.5.2

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

$- 6 - 2 x = 16 - 4 \sqrt{31 - 3 x} - 4 \sqrt{31 - 3 x} + {(31 - 3 x)}^{\frac{1}{2} \cdot 2}$

Step 4.3.1.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$- 6 - 2 x = 16 - 4 \sqrt{31 - 3 x} - 4 \sqrt{31 - 3 x} + {(31 - 3 x)}^{\frac{2}{2}}$

Step 4.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 4.3.1.3.1.5.4.1

Cancel the common factor.

$- 6 - 2 x = 16 - 4 \sqrt{31 - 3 x} - 4 \sqrt{31 - 3 x} + {(31 - 3 x)}^{\frac{2}{2}}$

Step 4.3.1.3.1.5.4.2

Rewrite the expression.

$- 6 - 2 x = 16 - 4 \sqrt{31 - 3 x} - 4 \sqrt{31 - 3 x} + {(31 - 3 x)}^{1}$

Step 4.3.1.3.1.5.5

Simplify.

$- 6 - 2 x = 16 - 4 \sqrt{31 - 3 x} - 4 \sqrt{31 - 3 x} + 31 - 3 x$

Step 4.3.1.3.2

Add $16$ and $31$ .

$- 6 - 2 x = 47 - 4 \sqrt{31 - 3 x} - 4 \sqrt{31 - 3 x} - 3 x$

Step 4.3.1.3.3

Subtract $4 \sqrt{31 - 3 x}$ from $- 4 \sqrt{31 - 3 x}$ .

$- 6 - 2 x = 47 - 8 \sqrt{31 - 3 x} - 3 x$

Step 5

Solve for $- 8 \sqrt{31 - 3 x}$ .

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Step 5.1

Rewrite the equation as $47 - 8 \sqrt{31 - 3 x} - 3 x = - 6 - 2 x$ .

$47 - 8 \sqrt{31 - 3 x} - 3 x = - 6 - 2 x$

Step 5.2

Move all terms not containing $- 8 \sqrt{31 - 3 x}$ to the right side of the equation.

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Step 5.2.1

Subtract $47$ from both sides of the equation.

$- 8 \sqrt{31 - 3 x} - 3 x = - 6 - 2 x - 47$

Step 5.2.2

Add $3 x$ to both sides of the equation.

$- 8 \sqrt{31 - 3 x} = - 6 - 2 x - 47 + 3 x$

Step 5.2.3

Subtract $47$ from $- 6$ .

$- 8 \sqrt{31 - 3 x} = - 2 x - 53 + 3 x$

Step 5.2.4

Add $- 2 x$ and $3 x$ .

$- 8 \sqrt{31 - 3 x} = x - 53$

Step 6

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

${(- 8 \sqrt{31 - 3 x})}^{2} = {(x - 53)}^{2}$

Step 7

Simplify each side of the equation.

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Step 7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{31 - 3 x}$ as ${(31 - 3 x)}^{\frac{1}{2}}$ .

${(- 8 {(31 - 3 x)}^{\frac{1}{2}})}^{2} = {(x - 53)}^{2}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Simplify ${(- 8 {(31 - 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 7.2.1.1

Apply the product rule to $- 8 {(31 - 3 x)}^{\frac{1}{2}}$ .

${(- 8)}^{2} {({(31 - 3 x)}^{\frac{1}{2}})}^{2} = {(x - 53)}^{2}$

Step 7.2.1.2

Raise $- 8$ to the power of $2$ .

$64 {({(31 - 3 x)}^{\frac{1}{2}})}^{2} = {(x - 53)}^{2}$

Step 7.2.1.3

Multiply the exponents in ${({(31 - 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 7.2.1.3.1

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

$64 {(31 - 3 x)}^{\frac{1}{2} \cdot 2} = {(x - 53)}^{2}$

Step 7.2.1.3.2

Cancel the common factor of $2$ .

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Step 7.2.1.3.2.1

Cancel the common factor.

$64 {(31 - 3 x)}^{\frac{1}{2} \cdot 2} = {(x - 53)}^{2}$

Step 7.2.1.3.2.2

Rewrite the expression.

$64 {(31 - 3 x)}^{1} = {(x - 53)}^{2}$

Step 7.2.1.4

Simplify.

$64 (31 - 3 x) = {(x - 53)}^{2}$

Step 7.2.1.5

Apply the distributive property.

$64 \cdot 31 + 64 (- 3 x) = {(x - 53)}^{2}$

Step 7.2.1.6

Multiply.

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Step 7.2.1.6.1

Multiply $64$ by $31$ .

$1984 + 64 (- 3 x) = {(x - 53)}^{2}$

Step 7.2.1.6.2

Multiply $- 3$ by $64$ .

$1984 - 192 x = {(x - 53)}^{2}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify ${(x - 53)}^{2}$ .

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Step 7.3.1.1

Rewrite ${(x - 53)}^{2}$ as $(x - 53) (x - 53)$ .

$1984 - 192 x = (x - 53) (x - 53)$

Step 7.3.1.2

Expand $(x - 53) (x - 53)$ using the FOIL Method.

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Step 7.3.1.2.1

Apply the distributive property.

$1984 - 192 x = x (x - 53) - 53 (x - 53)$

Step 7.3.1.2.2

Apply the distributive property.

$1984 - 192 x = x \cdot x + x \cdot - 53 - 53 (x - 53)$

Step 7.3.1.2.3

Apply the distributive property.

$1984 - 192 x = x \cdot x + x \cdot - 53 - 53 x - 53 \cdot - 53$

Step 7.3.1.3

Simplify and combine like terms.

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Step 7.3.1.3.1

Simplify each term.

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Step 7.3.1.3.1.1

Multiply $x$ by $x$ .

$1984 - 192 x = x^{2} + x \cdot - 53 - 53 x - 53 \cdot - 53$

Step 7.3.1.3.1.2

Move $- 53$ to the left of $x$ .

$1984 - 192 x = x^{2} - 53 \cdot x - 53 x - 53 \cdot - 53$

Step 7.3.1.3.1.3

Multiply $- 53$ by $- 53$ .

$1984 - 192 x = x^{2} - 53 x - 53 x + 2809$

Step 7.3.1.3.2

Subtract $53 x$ from $- 53 x$ .

$1984 - 192 x = x^{2} - 106 x + 2809$

Step 8

Solve for $x$ .

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Step 8.1

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

$x^{2} - 106 x + 2809 = 1984 - 192 x$

Step 8.2

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

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Step 8.2.1

Add $192 x$ to both sides of the equation.

$x^{2} - 106 x + 2809 + 192 x = 1984$

Step 8.2.2

Add $- 106 x$ and $192 x$ .

$x^{2} + 86 x + 2809 = 1984$

Step 8.3

Subtract $1984$ from both sides of the equation.

$x^{2} + 86 x + 2809 - 1984 = 0$

Step 8.4

Subtract $1984$ from $2809$ .

$x^{2} + 86 x + 825 = 0$

Step 8.5

Factor $x^{2} + 86 x + 825$ using the AC method.

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Step 8.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $825$ and whose sum is $86$ .

$11, 75$

Step 8.5.2

Write the factored form using these integers.

$(x + 11) (x + 75) = 0$

Step 8.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 11 = 0$

$x + 75 = 0$

Step 8.7

Set $x + 11$ equal to $0$ and solve for $x$ .

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Step 8.7.1

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

Step 8.7.2

Subtract $11$ from both sides of the equation.

$x = - 11$

Step 8.8

Set $x + 75$ equal to $0$ and solve for $x$ .

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Step 8.8.1

Set $x + 75$ equal to $0$ .

$x + 75 = 0$

Step 8.8.2

Subtract $75$ from both sides of the equation.

$x = - 75$

Step 8.9

The final solution is all the values that make $(x + 11) (x + 75) = 0$ true.

$x = - 11, - 75$

Step 9

Exclude the solutions that do not make $4 = \sqrt{- 6 - 2 x} + \sqrt{31 - 3 x}$ true.

$No solution$