Algebra Examples

$\sqrt{6 + 5 x} + \sqrt{3 x + 4} = 2$

Step 1

Subtract $\sqrt{3 x + 4}$ from both sides of the equation.

$\sqrt{6 + 5 x} = 2 - \sqrt{3 x + 4}$

Step 2

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

${\sqrt{6 + 5 x}}^{2} = {(2 - \sqrt{3 x + 4})}^{2}$

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 $\sqrt{6 + 5 x}$ as ${(6 + 5 x)}^{\frac{1}{2}}$ .

${({(6 + 5 x)}^{\frac{1}{2}})}^{2} = {(2 - \sqrt{3 x + 4})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(6 + 5 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(6 + 5 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(6 + 5 x)}^{\frac{1}{2} \cdot 2} = {(2 - \sqrt{3 x + 4})}^{2}$

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.

${(6 + 5 x)}^{\frac{1}{2} \cdot 2} = {(2 - \sqrt{3 x + 4})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(6 + 5 x)}^{1} = {(2 - \sqrt{3 x + 4})}^{2}$

Step 3.2.1.2

Simplify.

$6 + 5 x = {(2 - \sqrt{3 x + 4})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

$6 + 5 x = (2 - \sqrt{3 x + 4}) (2 - \sqrt{3 x + 4})$

Step 3.3.1.2

Expand $(2 - \sqrt{3 x + 4}) (2 - \sqrt{3 x + 4})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$6 + 5 x = 2 (2 - \sqrt{3 x + 4}) - \sqrt{3 x + 4} (2 - \sqrt{3 x + 4})$

Step 3.3.1.2.2

Apply the distributive property.

$6 + 5 x = 2 \cdot 2 + 2 (- \sqrt{3 x + 4}) - \sqrt{3 x + 4} (2 - \sqrt{3 x + 4})$

Step 3.3.1.2.3

Apply the distributive property.

$6 + 5 x = 2 \cdot 2 + 2 (- \sqrt{3 x + 4}) - \sqrt{3 x + 4} \cdot 2 - \sqrt{3 x + 4} (- \sqrt{3 x + 4})$

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 $2$ by $2$ .

$6 + 5 x = 4 + 2 (- \sqrt{3 x + 4}) - \sqrt{3 x + 4} \cdot 2 - \sqrt{3 x + 4} (- \sqrt{3 x + 4})$

Step 3.3.1.3.1.2

Multiply $- 1$ by $2$ .

$6 + 5 x = 4 - 2 \sqrt{3 x + 4} - \sqrt{3 x + 4} \cdot 2 - \sqrt{3 x + 4} (- \sqrt{3 x + 4})$

Step 3.3.1.3.1.3

Multiply $2$ by $- 1$ .

$6 + 5 x = 4 - 2 \sqrt{3 x + 4} - 2 \sqrt{3 x + 4} - \sqrt{3 x + 4} (- \sqrt{3 x + 4})$

Step 3.3.1.3.1.4

Multiply $- \sqrt{3 x + 4} (- \sqrt{3 x + 4})$ .

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Step 3.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$6 + 5 x = 4 - 2 \sqrt{3 x + 4} - 2 \sqrt{3 x + 4} + 1 \sqrt{3 x + 4} \sqrt{3 x + 4}$

Step 3.3.1.3.1.4.2

Multiply $\sqrt{3 x + 4}$ by $1$ .

$6 + 5 x = 4 - 2 \sqrt{3 x + 4} - 2 \sqrt{3 x + 4} + \sqrt{3 x + 4} \sqrt{3 x + 4}$

Step 3.3.1.3.1.4.3

Raise $\sqrt{3 x + 4}$ to the power of $1$ .

$6 + 5 x = 4 - 2 \sqrt{3 x + 4} - 2 \sqrt{3 x + 4} + {\sqrt{3 x + 4}}^{1} \sqrt{3 x + 4}$

Step 3.3.1.3.1.4.4

Raise $\sqrt{3 x + 4}$ to the power of $1$ .

$6 + 5 x = 4 - 2 \sqrt{3 x + 4} - 2 \sqrt{3 x + 4} + {\sqrt{3 x + 4}}^{1} {\sqrt{3 x + 4}}^{1}$

Step 3.3.1.3.1.4.5

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

$6 + 5 x = 4 - 2 \sqrt{3 x + 4} - 2 \sqrt{3 x + 4} + {\sqrt{3 x + 4}}^{1 + 1}$

Step 3.3.1.3.1.4.6

Add $1$ and $1$ .

$6 + 5 x = 4 - 2 \sqrt{3 x + 4} - 2 \sqrt{3 x + 4} + {\sqrt{3 x + 4}}^{2}$

Step 3.3.1.3.1.5

Rewrite ${\sqrt{3 x + 4}}^{2}$ as $3 x + 4$ .

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Step 3.3.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 4}$ as ${(3 x + 4)}^{\frac{1}{2}}$ .

$6 + 5 x = 4 - 2 \sqrt{3 x + 4} - 2 \sqrt{3 x + 4} + {({(3 x + 4)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.5.2

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

$6 + 5 x = 4 - 2 \sqrt{3 x + 4} - 2 \sqrt{3 x + 4} + {(3 x + 4)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.5.3

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

$6 + 5 x = 4 - 2 \sqrt{3 x + 4} - 2 \sqrt{3 x + 4} + {(3 x + 4)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.5.4.1

Cancel the common factor.

$6 + 5 x = 4 - 2 \sqrt{3 x + 4} - 2 \sqrt{3 x + 4} + {(3 x + 4)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

$6 + 5 x = 4 - 2 \sqrt{3 x + 4} - 2 \sqrt{3 x + 4} + {(3 x + 4)}^{1}$

Step 3.3.1.3.1.5.5

Simplify.

$6 + 5 x = 4 - 2 \sqrt{3 x + 4} - 2 \sqrt{3 x + 4} + 3 x + 4$

Step 3.3.1.3.2

Add $4$ and $4$ .

$6 + 5 x = 8 - 2 \sqrt{3 x + 4} - 2 \sqrt{3 x + 4} + 3 x$

Step 3.3.1.3.3

Subtract $2 \sqrt{3 x + 4}$ from $- 2 \sqrt{3 x + 4}$ .

$6 + 5 x = 8 - 4 \sqrt{3 x + 4} + 3 x$

Step 4

Solve for $- 4 \sqrt{3 x + 4}$ .

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Step 4.1

Rewrite the equation as $8 - 4 \sqrt{3 x + 4} + 3 x = 6 + 5 x$ .

$8 - 4 \sqrt{3 x + 4} + 3 x = 6 + 5 x$

Step 4.2

Move all terms not containing $- 4 \sqrt{3 x + 4}$ to the right side of the equation.

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Step 4.2.1

Subtract $8$ from both sides of the equation.

$- 4 \sqrt{3 x + 4} + 3 x = 6 + 5 x - 8$

Step 4.2.2

Subtract $3 x$ from both sides of the equation.

$- 4 \sqrt{3 x + 4} = 6 + 5 x - 8 - 3 x$

Step 4.2.3

Subtract $8$ from $6$ .

$- 4 \sqrt{3 x + 4} = 5 x - 2 - 3 x$

Step 4.2.4

Subtract $3 x$ from $5 x$ .

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

Step 5

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

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

Step 6

Simplify each side of the equation.

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 4}$ as ${(3 x + 4)}^{\frac{1}{2}}$ .

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(- 4 {(3 x + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $- 4 {(3 x + 4)}^{\frac{1}{2}}$ .

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

Step 6.2.1.2

Raise $- 4$ to the power of $2$ .

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

Step 6.2.1.3

Multiply the exponents in ${({(3 x + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

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

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

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

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

Step 6.2.1.3.2.2

Rewrite the expression.

$16 {(3 x + 4)}^{1} = {(2 x - 2)}^{2}$

Step 6.2.1.4

Simplify.

$16 (3 x + 4) = {(2 x - 2)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$16 (3 x) + 16 \cdot 4 = {(2 x - 2)}^{2}$

Step 6.2.1.6

Multiply.

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Step 6.2.1.6.1

Multiply $3$ by $16$ .

$48 x + 16 \cdot 4 = {(2 x - 2)}^{2}$

Step 6.2.1.6.2

Multiply $16$ by $4$ .

$48 x + 64 = {(2 x - 2)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

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

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Step 6.3.1.1

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

$48 x + 64 = (2 x - 2) (2 x - 2)$

Step 6.3.1.2

Expand $(2 x - 2) (2 x - 2)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$48 x + 64 = 2 x (2 x - 2) - 2 (2 x - 2)$

Step 6.3.1.2.2

Apply the distributive property.

$48 x + 64 = 2 x (2 x) + 2 x \cdot - 2 - 2 (2 x - 2)$

Step 6.3.1.2.3

Apply the distributive property.

$48 x + 64 = 2 x (2 x) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

Step 6.3.1.3

Simplify and combine like terms.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$48 x + 64 = 2 \cdot 2 x \cdot x + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

Step 6.3.1.3.1.2

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

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Step 6.3.1.3.1.2.1

Move $x$ .

$48 x + 64 = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

Step 6.3.1.3.1.2.2

Multiply $x$ by $x$ .

$48 x + 64 = 2 \cdot 2 x^{2} + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

Step 6.3.1.3.1.3

Multiply $2$ by $2$ .

$48 x + 64 = 4 x^{2} + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

Step 6.3.1.3.1.4

Multiply $- 2$ by $2$ .

$48 x + 64 = 4 x^{2} - 4 x - 2 (2 x) - 2 \cdot - 2$

Step 6.3.1.3.1.5

Multiply $2$ by $- 2$ .

$48 x + 64 = 4 x^{2} - 4 x - 4 x - 2 \cdot - 2$

Step 6.3.1.3.1.6

Multiply $- 2$ by $- 2$ .

$48 x + 64 = 4 x^{2} - 4 x - 4 x + 4$

Step 6.3.1.3.2

Subtract $4 x$ from $- 4 x$ .

$48 x + 64 = 4 x^{2} - 8 x + 4$

Step 7

Solve for $x$ .

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Step 7.1

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

$4 x^{2} - 8 x + 4 = 48 x + 64$

Step 7.2

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

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Step 7.2.1

Subtract $48 x$ from both sides of the equation.

$4 x^{2} - 8 x + 4 - 48 x = 64$

Step 7.2.2

Subtract $48 x$ from $- 8 x$ .

$4 x^{2} - 56 x + 4 = 64$

Step 7.3

Subtract $64$ from both sides of the equation.

$4 x^{2} - 56 x + 4 - 64 = 0$

Step 7.4

Subtract $64$ from $4$ .

$4 x^{2} - 56 x - 60 = 0$

Step 7.5

Factor the left side of the equation.

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Step 7.5.1

Factor $4$ out of $4 x^{2} - 56 x - 60$ .

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Step 7.5.1.1

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

$4 (x^{2}) - 56 x - 60 = 0$

Step 7.5.1.2

Factor $4$ out of $- 56 x$ .

$4 (x^{2}) + 4 (- 14 x) - 60 = 0$

Step 7.5.1.3

Factor $4$ out of $- 60$ .

$4 x^{2} + 4 (- 14 x) + 4 \cdot - 15 = 0$

Step 7.5.1.4

Factor $4$ out of $4 x^{2} + 4 (- 14 x)$ .

$4 (x^{2} - 14 x) + 4 \cdot - 15 = 0$

Step 7.5.1.5

Factor $4$ out of $4 (x^{2} - 14 x) + 4 \cdot - 15$ .

$4 (x^{2} - 14 x - 15) = 0$

Step 7.5.2

Factor.

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Step 7.5.2.1

Factor $x^{2} - 14 x - 15$ using the AC method.

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Step 7.5.2.1.1

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 $- 15$ and whose sum is $- 14$ .

$- 15, 1$

Step 7.5.2.1.2

Write the factored form using these integers.

$4 ((x - 15) (x + 1)) = 0$

Step 7.5.2.2

Remove unnecessary parentheses.

$4 (x - 15) (x + 1) = 0$

Step 7.6

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

$x - 15 = 0$

$x + 1 = 0$

Step 7.7

Set $x - 15$ equal to $0$ and solve for $x$ .

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Step 7.7.1

Set $x - 15$ equal to $0$ .

$x - 15 = 0$

Step 7.7.2

Add $15$ to both sides of the equation.

$x = 15$

Step 7.8

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 7.8.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 7.8.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7.9

The final solution is all the values that make $4 (x - 15) (x + 1) = 0$ true.

$x = 15, - 1$

Step 8

Exclude the solutions that do not make $\sqrt{6 + 5 x} + \sqrt{3 x + 4} = 2$ true.

$x = - 1$