Pre-Algebra Examples

$\sqrt{6 x + 7} - \sqrt{3 x + 3} = 1$

Step 1

Solve for $\sqrt{6 x + 7}$ .

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Step 1.1

Factor $3$ out of $3 x + 3$ .

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Step 1.1.1

Factor $3$ out of $3 x$ .

$\sqrt{6 x + 7} - \sqrt{3 (x) + 3} = 1$

Step 1.1.2

Factor $3$ out of $3$ .

$\sqrt{6 x + 7} - \sqrt{3 (x) + 3 (1)} = 1$

Step 1.1.3

Factor $3$ out of $3 (x) + 3 (1)$ .

$\sqrt{6 x + 7} - \sqrt{3 (x + 1)} = 1$

Step 1.2

Add $\sqrt{3 (x + 1)}$ to both sides of the equation.

$\sqrt{6 x + 7} = 1 + \sqrt{3 (x + 1)}$

Step 2

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

${\sqrt{6 x + 7}}^{2} = {(1 + \sqrt{3 (x + 1)})}^{2}$

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 $\sqrt{6 x + 7}$ as ${(6 x + 7)}^{\frac{1}{2}}$ .

${({(6 x + 7)}^{\frac{1}{2}})}^{2} = {(1 + \sqrt{3 (x + 1)})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(6 x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(6 x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(6 x + 7)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{3 (x + 1)})}^{2}$

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.

${(6 x + 7)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{3 (x + 1)})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(6 x + 7)}^{1} = {(1 + \sqrt{3 (x + 1)})}^{2}$

Step 3.2.1.2

Simplify.

$6 x + 7 = {(1 + \sqrt{3 (x + 1)})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(1 + \sqrt{3 (x + 1)})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(1 + \sqrt{3 (x + 1)})}^{2}$ as $(1 + \sqrt{3 (x + 1)}) (1 + \sqrt{3 (x + 1)})$ .

$6 x + 7 = (1 + \sqrt{3 (x + 1)}) (1 + \sqrt{3 (x + 1)})$

Step 3.3.1.2

Expand $(1 + \sqrt{3 (x + 1)}) (1 + \sqrt{3 (x + 1)})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$6 x + 7 = 1 (1 + \sqrt{3 (x + 1)}) + \sqrt{3 (x + 1)} (1 + \sqrt{3 (x + 1)})$

Step 3.3.1.2.2

Apply the distributive property.

$6 x + 7 = 1 \cdot 1 + 1 \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} (1 + \sqrt{3 (x + 1)})$

Step 3.3.1.2.3

Apply the distributive property.

$6 x + 7 = 1 \cdot 1 + 1 \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} \cdot 1 + \sqrt{3 (x + 1)} \sqrt{3 (x + 1)}$

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 $1$ by $1$ .

$6 x + 7 = 1 + 1 \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} \cdot 1 + \sqrt{3 (x + 1)} \sqrt{3 (x + 1)}$

Step 3.3.1.3.1.2

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

$6 x + 7 = 1 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} \cdot 1 + \sqrt{3 (x + 1)} \sqrt{3 (x + 1)}$

Step 3.3.1.3.1.3

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

$6 x + 7 = 1 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} \sqrt{3 (x + 1)}$

Step 3.3.1.3.1.4

Multiply $\sqrt{3 (x + 1)} \sqrt{3 (x + 1)}$ .

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Step 3.3.1.3.1.4.1

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

$6 x + 7 = 1 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} + {\sqrt{3 (x + 1)}}^{1} \sqrt{3 (x + 1)}$

Step 3.3.1.3.1.4.2

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

$6 x + 7 = 1 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} + {\sqrt{3 (x + 1)}}^{1} {\sqrt{3 (x + 1)}}^{1}$

Step 3.3.1.3.1.4.3

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

$6 x + 7 = 1 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} + {\sqrt{3 (x + 1)}}^{1 + 1}$

Step 3.3.1.3.1.4.4

Add $1$ and $1$ .

$6 x + 7 = 1 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} + {\sqrt{3 (x + 1)}}^{2}$

Step 3.3.1.3.1.5

Rewrite ${\sqrt{3 (x + 1)}}^{2}$ as $3 (x + 1)$ .

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Step 3.3.1.3.1.5.1

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

$6 x + 7 = 1 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} + {({(3 (x + 1))}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.5.2

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

$6 x + 7 = 1 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} + {(3 (x + 1))}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.5.3

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

$6 x + 7 = 1 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} + {(3 (x + 1))}^{\frac{2}{2}}$

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.

$6 x + 7 = 1 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} + {(3 (x + 1))}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

$6 x + 7 = 1 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} + {(3 (x + 1))}^{1}$

Step 3.3.1.3.1.5.5

Simplify.

$6 x + 7 = 1 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} + 3 (x + 1)$

Step 3.3.1.3.1.6

Apply the distributive property.

$6 x + 7 = 1 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} + 3 x + 3 \cdot 1$

Step 3.3.1.3.1.7

Multiply $3$ by $1$ .

$6 x + 7 = 1 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} + 3 x + 3$

Step 3.3.1.3.2

Add $1$ and $3$ .

$6 x + 7 = 4 + \sqrt{3 (x + 1)} + \sqrt{3 (x + 1)} + 3 x$

Step 3.3.1.3.3

Add $\sqrt{3 (x + 1)}$ and $\sqrt{3 (x + 1)}$ .

$6 x + 7 = 4 + 2 \sqrt{3 (x + 1)} + 3 x$

Step 4

Solve for $2 \sqrt{3 (x + 1)}$ .

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Step 4.1

Rewrite the equation as $4 + 2 \sqrt{3 (x + 1)} + 3 x = 6 x + 7$ .

$4 + 2 \sqrt{3 (x + 1)} + 3 x = 6 x + 7$

Step 4.2

Move all terms not containing $2 \sqrt{3 (x + 1)}$ to the right side of the equation.

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Step 4.2.1

Subtract $4$ from both sides of the equation.

$2 \sqrt{3 (x + 1)} + 3 x = 6 x + 7 - 4$

Step 4.2.2

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

$2 \sqrt{3 (x + 1)} = 6 x + 7 - 4 - 3 x$

Step 4.2.3

Subtract $3 x$ from $6 x$ .

$2 \sqrt{3 (x + 1)} = 3 x + 7 - 4$

Step 4.2.4

Subtract $4$ from $7$ .

$2 \sqrt{3 (x + 1)} = 3 x + 3$

Step 5

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

${(2 \sqrt{3 (x + 1)})}^{2} = {(3 x + 3)}^{2}$

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 $\sqrt{3 (x + 1)}$ as ${(3 (x + 1))}^{\frac{1}{2}}$ .

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

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

Apply the distributive property.

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

Step 6.2.1.2

Simplify the expression.

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Step 6.2.1.2.1

Multiply $3$ by $1$ .

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

Step 6.2.1.2.2

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

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

Step 6.2.1.2.3

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

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

Step 6.2.1.2.4

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

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Step 6.2.1.2.4.1

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

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

Step 6.2.1.2.4.2

Cancel the common factor of $2$ .

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Step 6.2.1.2.4.2.1

Cancel the common factor.

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

Step 6.2.1.2.4.2.2

Rewrite the expression.

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

Step 6.2.1.3

Simplify.

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

Step 6.2.1.4

Apply the distributive property.

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

Step 6.2.1.5

Multiply.

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Step 6.2.1.5.1

Multiply $3$ by $4$ .

$12 x + 4 \cdot 3 = {(3 x + 3)}^{2}$

Step 6.2.1.5.2

Multiply $4$ by $3$ .

$12 x + 12 = {(3 x + 3)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(3 x + 3)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(3 x + 3)}^{2}$ as $(3 x + 3) (3 x + 3)$ .

$12 x + 12 = (3 x + 3) (3 x + 3)$

Step 6.3.1.2

Expand $(3 x + 3) (3 x + 3)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$12 x + 12 = 3 x (3 x + 3) + 3 (3 x + 3)$

Step 6.3.1.2.2

Apply the distributive property.

$12 x + 12 = 3 x (3 x) + 3 x \cdot 3 + 3 (3 x + 3)$

Step 6.3.1.2.3

Apply the distributive property.

$12 x + 12 = 3 x (3 x) + 3 x \cdot 3 + 3 (3 x) + 3 \cdot 3$

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.

$12 x + 12 = 3 \cdot 3 x \cdot x + 3 x \cdot 3 + 3 (3 x) + 3 \cdot 3$

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 $x$ .

$12 x + 12 = 3 \cdot 3 (x \cdot x) + 3 x \cdot 3 + 3 (3 x) + 3 \cdot 3$

Step 6.3.1.3.1.2.2

Multiply $x$ by $x$ .

$12 x + 12 = 3 \cdot 3 x^{2} + 3 x \cdot 3 + 3 (3 x) + 3 \cdot 3$

Step 6.3.1.3.1.3

Multiply $3$ by $3$ .

$12 x + 12 = 9 x^{2} + 3 x \cdot 3 + 3 (3 x) + 3 \cdot 3$

Step 6.3.1.3.1.4

Multiply $3$ by $3$ .

$12 x + 12 = 9 x^{2} + 9 x + 3 (3 x) + 3 \cdot 3$

Step 6.3.1.3.1.5

Multiply $3$ by $3$ .

$12 x + 12 = 9 x^{2} + 9 x + 9 x + 3 \cdot 3$

Step 6.3.1.3.1.6

Multiply $3$ by $3$ .

$12 x + 12 = 9 x^{2} + 9 x + 9 x + 9$

Step 6.3.1.3.2

Add $9 x$ and $9 x$ .

$12 x + 12 = 9 x^{2} + 18 x + 9$

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.

$9 x^{2} + 18 x + 9 = 12 x + 12$

Step 7.2

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

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Step 7.2.1

Subtract $12 x$ from both sides of the equation.

$9 x^{2} + 18 x + 9 - 12 x = 12$

Step 7.2.2

Subtract $12 x$ from $18 x$ .

$9 x^{2} + 6 x + 9 = 12$

Step 7.3

Subtract $12$ from both sides of the equation.

$9 x^{2} + 6 x + 9 - 12 = 0$

Step 7.4

Subtract $12$ from $9$ .

$9 x^{2} + 6 x - 3 = 0$

Step 7.5

Factor the left side of the equation.

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Step 7.5.1

Factor $3$ out of $9 x^{2} + 6 x - 3$ .

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Step 7.5.1.1

Factor $3$ out of $9 x^{2}$ .

$3 (3 x^{2}) + 6 x - 3 = 0$

Step 7.5.1.2

Factor $3$ out of $6 x$ .

$3 (3 x^{2}) + 3 (2 x) - 3 = 0$

Step 7.5.1.3

Factor $3$ out of $- 3$ .

$3 (3 x^{2}) + 3 (2 x) + 3 (- 1) = 0$

Step 7.5.1.4

Factor $3$ out of $3 (3 x^{2}) + 3 (2 x)$ .

$3 (3 x^{2} + 2 x) + 3 (- 1) = 0$

Step 7.5.1.5

Factor $3$ out of $3 (3 x^{2} + 2 x) + 3 (- 1)$ .

$3 (3 x^{2} + 2 x - 1) = 0$

Step 7.5.2

Factor.

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Step 7.5.2.1

Factor by grouping.

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Step 7.5.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $b = 2$ .

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Step 7.5.2.1.1.1

Factor $2$ out of $2 x$ .

$3 (3 x^{2} + 2 (x) - 1) = 0$

Step 7.5.2.1.1.2

Rewrite $2$ as $- 1$ plus $3$

$3 (3 x^{2} + (- 1 + 3) x - 1) = 0$

Step 7.5.2.1.1.3

Apply the distributive property.

$3 (3 x^{2} - 1 x + 3 x - 1) = 0$

Step 7.5.2.1.2

Factor out the greatest common factor from each group.

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Step 7.5.2.1.2.1

Group the first two terms and the last two terms.

$3 ((3 x^{2} - 1 x) + 3 x - 1) = 0$

Step 7.5.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$3 (x (3 x - 1) + 1 (3 x - 1)) = 0$

Step 7.5.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$3 ((3 x - 1) (x + 1)) = 0$

Step 7.5.2.2

Remove unnecessary parentheses.

$3 (3 x - 1) (x + 1) = 0$

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 $0$ .

$3 x - 1 = 0$

$x + 1 = 0$

Step 7.7

Set $3 x - 1$ equal to $0$ and solve for $x$ .

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Step 7.7.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 7.7.2

Solve $3 x - 1 = 0$ for $x$ .

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Step 7.7.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 7.7.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 7.7.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Step 7.7.2.2.2

Simplify the left side.

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Step 7.7.2.2.2.1

Cancel the common factor of $3$ .

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Step 7.7.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 7.7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

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 $3 (3 x - 1) (x + 1) = 0$ true.

$x = \frac{1}{3}, - 1$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{3}, - 1$

Decimal Form:

$x = 0.\overline{3}, - 1$