Algebra Examples

$\sqrt{3 x^{2}} - x \sqrt{12} + 2 x \sqrt{75} = \sqrt{3}$

Step 1

Solve for $\sqrt{3 x^{2}}$ .

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Step 1.1

Simplify $\sqrt{3 x^{2}} - x \sqrt{12} + 2 x \sqrt{75}$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.1.1.1.1

Factor $4$ out of $12$ .

$\sqrt{3 x^{2}} - x \sqrt{4 (3)} + 2 x \sqrt{75} = \sqrt{3}$

Step 1.1.1.1.2

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

$\sqrt{3 x^{2}} - x \sqrt{2^{2} \cdot 3} + 2 x \sqrt{75} = \sqrt{3}$

Step 1.1.1.2

Pull terms out from under the radical.

$\sqrt{3 x^{2}} - x (2 \sqrt{3}) + 2 x \sqrt{75} = \sqrt{3}$

Step 1.1.1.3

Multiply $2$ by $- 1$ .

$\sqrt{3 x^{2}} - 2 x \sqrt{3} + 2 x \sqrt{75} = \sqrt{3}$

Step 1.1.1.4

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 1.1.1.4.1

Factor $25$ out of $75$ .

$\sqrt{3 x^{2}} - 2 x \sqrt{3} + 2 x \sqrt{25 (3)} = \sqrt{3}$

Step 1.1.1.4.2

Rewrite $25$ as $5^{2}$ .

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

Step 1.1.1.5

Pull terms out from under the radical.

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

Step 1.1.1.6

Multiply $5$ by $2$ .

$\sqrt{3 x^{2}} - 2 x \sqrt{3} + 10 x \sqrt{3} = \sqrt{3}$

Step 1.1.2

Add $- 2 x \sqrt{3}$ and $10 x \sqrt{3}$ .

$\sqrt{3 x^{2}} + 8 x \sqrt{3} = \sqrt{3}$

Step 1.2

Subtract $8 x \sqrt{3}$ from both sides of the equation.

$\sqrt{3 x^{2}} = \sqrt{3} - 8 x \sqrt{3}$

Step 2

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

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

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

${(3 x^{2})}^{\frac{1}{2} \cdot 2} = {(\sqrt{3} - 8 x \sqrt{3})}^{2}$

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.

${(3 x^{2})}^{\frac{1}{2} \cdot 2} = {(\sqrt{3} - 8 x \sqrt{3})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(3 x^{2})}^{1} = {(\sqrt{3} - 8 x \sqrt{3})}^{2}$

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(\sqrt{3} - 8 x \sqrt{3})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(\sqrt{3} - 8 x \sqrt{3})}^{2}$ as $(\sqrt{3} - 8 x \sqrt{3}) (\sqrt{3} - 8 x \sqrt{3})$ .

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

Step 3.3.1.2

Expand $(\sqrt{3} - 8 x \sqrt{3}) (\sqrt{3} - 8 x \sqrt{3})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

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

Step 3.3.1.2.2

Apply the distributive property.

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

Step 3.3.1.2.3

Apply the distributive property.

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

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

Combine using the product rule for radicals.

$3 x^{2} = \sqrt{3 \cdot 3} + \sqrt{3} (- 8 x \sqrt{3}) - 8 x \sqrt{3} \sqrt{3} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.2

Multiply $3$ by $3$ .

$3 x^{2} = \sqrt{9} + \sqrt{3} (- 8 x \sqrt{3}) - 8 x \sqrt{3} \sqrt{3} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.3

Rewrite $9$ as $3^{2}$ .

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

Step 3.3.1.3.1.4

Pull terms out from under the radical, assuming positive real numbers.

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

Step 3.3.1.3.1.5

Multiply $\sqrt{3} (- 8 x \sqrt{3})$ .

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Step 3.3.1.3.1.5.1

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

$3 x^{2} = 3 - 8 x ({\sqrt{3}}^{1} \sqrt{3}) - 8 x \sqrt{3} \sqrt{3} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.5.2

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

$3 x^{2} = 3 - 8 x ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - 8 x \sqrt{3} \sqrt{3} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.5.3

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

$3 x^{2} = 3 - 8 x {\sqrt{3}}^{1 + 1} - 8 x \sqrt{3} \sqrt{3} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.5.4

Add $1$ and $1$ .

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

Step 3.3.1.3.1.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.3.1.3.1.6.1

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

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

Step 3.3.1.3.1.6.2

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

$3 x^{2} = 3 - 8 x \cdot 3^{\frac{1}{2} \cdot 2} - 8 x \sqrt{3} \sqrt{3} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.6.3

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

$3 x^{2} = 3 - 8 x \cdot 3^{\frac{2}{2}} - 8 x \sqrt{3} \sqrt{3} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.6.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.6.4.1

Cancel the common factor.

$3 x^{2} = 3 - 8 x \cdot 3^{\frac{2}{2}} - 8 x \sqrt{3} \sqrt{3} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.6.4.2

Rewrite the expression.

$3 x^{2} = 3 - 8 x \cdot 3^{1} - 8 x \sqrt{3} \sqrt{3} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.6.5

Evaluate the exponent.

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

Step 3.3.1.3.1.7

Multiply $3$ by $- 8$ .

$3 x^{2} = 3 - 24 x - 8 x \sqrt{3} \sqrt{3} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.8

Multiply $- 8 x \sqrt{3} \sqrt{3}$ .

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Step 3.3.1.3.1.8.1

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

$3 x^{2} = 3 - 24 x - 8 x ({\sqrt{3}}^{1} \sqrt{3}) - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.8.2

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

$3 x^{2} = 3 - 24 x - 8 x ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.8.3

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

$3 x^{2} = 3 - 24 x - 8 x {\sqrt{3}}^{1 + 1} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.8.4

Add $1$ and $1$ .

$3 x^{2} = 3 - 24 x - 8 x {\sqrt{3}}^{2} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.9

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.3.1.3.1.9.1

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

$3 x^{2} = 3 - 24 x - 8 x {(3^{\frac{1}{2}})}^{2} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.9.2

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

$3 x^{2} = 3 - 24 x - 8 x \cdot 3^{\frac{1}{2} \cdot 2} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.9.3

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

$3 x^{2} = 3 - 24 x - 8 x \cdot 3^{\frac{2}{2}} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.9.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.9.4.1

Cancel the common factor.

$3 x^{2} = 3 - 24 x - 8 x \cdot 3^{\frac{2}{2}} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.9.4.2

Rewrite the expression.

$3 x^{2} = 3 - 24 x - 8 x \cdot 3^{1} - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.9.5

Evaluate the exponent.

$3 x^{2} = 3 - 24 x - 8 x \cdot 3 - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.10

Multiply $3$ by $- 8$ .

$3 x^{2} = 3 - 24 x - 24 x - 8 x \sqrt{3} (- 8 x \sqrt{3})$

Step 3.3.1.3.1.11

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

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Step 3.3.1.3.1.11.1

Move $x$ .

$3 x^{2} = 3 - 24 x - 24 x - 8 (x \cdot x) \sqrt{3} (- 8 \sqrt{3})$

Step 3.3.1.3.1.11.2

Multiply $x$ by $x$ .

$3 x^{2} = 3 - 24 x - 24 x - 8 x^{2} \sqrt{3} (- 8 \sqrt{3})$

Step 3.3.1.3.1.12

Multiply $- 8 x^{2} \sqrt{3} (- 8 \sqrt{3})$ .

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Step 3.3.1.3.1.12.1

Multiply $- 8$ by $- 8$ .

$3 x^{2} = 3 - 24 x - 24 x + 64 x^{2} \sqrt{3} \sqrt{3}$

Step 3.3.1.3.1.12.2

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

$3 x^{2} = 3 - 24 x - 24 x + 64 x^{2} ({\sqrt{3}}^{1} \sqrt{3})$

Step 3.3.1.3.1.12.3

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

$3 x^{2} = 3 - 24 x - 24 x + 64 x^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 3.3.1.3.1.12.4

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

$3 x^{2} = 3 - 24 x - 24 x + 64 x^{2} {\sqrt{3}}^{1 + 1}$

Step 3.3.1.3.1.12.5

Add $1$ and $1$ .

$3 x^{2} = 3 - 24 x - 24 x + 64 x^{2} {\sqrt{3}}^{2}$

Step 3.3.1.3.1.13

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.3.1.3.1.13.1

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

$3 x^{2} = 3 - 24 x - 24 x + 64 x^{2} {(3^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.13.2

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

$3 x^{2} = 3 - 24 x - 24 x + 64 x^{2} \cdot 3^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.13.3

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

$3 x^{2} = 3 - 24 x - 24 x + 64 x^{2} \cdot 3^{\frac{2}{2}}$

Step 3.3.1.3.1.13.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.13.4.1

Cancel the common factor.

$3 x^{2} = 3 - 24 x - 24 x + 64 x^{2} \cdot 3^{\frac{2}{2}}$

Step 3.3.1.3.1.13.4.2

Rewrite the expression.

$3 x^{2} = 3 - 24 x - 24 x + 64 x^{2} \cdot 3^{1}$

Step 3.3.1.3.1.13.5

Evaluate the exponent.

$3 x^{2} = 3 - 24 x - 24 x + 64 x^{2} \cdot 3$

Step 3.3.1.3.1.14

Multiply $3$ by $64$ .

$3 x^{2} = 3 - 24 x - 24 x + 192 x^{2}$

Step 3.3.1.3.2

Subtract $24 x$ from $- 24 x$ .

$3 x^{2} = 3 - 48 x + 192 x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

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

$3 - 48 x + 192 x^{2} = 3 x^{2}$

Step 4.2

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

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Step 4.2.1

Subtract $3 x^{2}$ from both sides of the equation.

$3 - 48 x + 192 x^{2} - 3 x^{2} = 0$

Step 4.2.2

Subtract $3 x^{2}$ from $192 x^{2}$ .

$3 - 48 x + 189 x^{2} = 0$

Step 4.3

Factor the left side of the equation.

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Step 4.3.1

Factor $3$ out of $3 - 48 x + 189 x^{2}$ .

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Step 4.3.1.1

Factor $3$ out of $3$ .

$3 (1) - 48 x + 189 x^{2} = 0$

Step 4.3.1.2

Factor $3$ out of $- 48 x$ .

$3 (1) + 3 (- 16 x) + 189 x^{2} = 0$

Step 4.3.1.3

Factor $3$ out of $189 x^{2}$ .

$3 (1) + 3 (- 16 x) + 3 (63 x^{2}) = 0$

Step 4.3.1.4

Factor $3$ out of $3 (1) + 3 (- 16 x)$ .

$3 (1 - 16 x) + 3 (63 x^{2}) = 0$

Step 4.3.1.5

Factor $3$ out of $3 (1 - 16 x) + 3 (63 x^{2})$ .

$3 (1 - 16 x + 63 x^{2}) = 0$

Step 4.3.2

Factor.

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Step 4.3.2.1

Factor by grouping.

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Step 4.3.2.1.1

Reorder terms.

$3 (63 x^{2} - 16 x + 1) = 0$

Step 4.3.2.1.2

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 $a \cdot c = 63 \cdot 1 = 63$ and whose sum is $b = - 16$ .

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Step 4.3.2.1.2.1

Factor $- 16$ out of $- 16 x$ .

$3 (63 x^{2} - 16 x + 1) = 0$

Step 4.3.2.1.2.2

Rewrite $- 16$ as $- 7$ plus $- 9$

$3 (63 x^{2} + (- 7 - 9) x + 1) = 0$

Step 4.3.2.1.2.3

Apply the distributive property.

$3 (63 x^{2} - 7 x - 9 x + 1) = 0$

Step 4.3.2.1.3

Factor out the greatest common factor from each group.

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Step 4.3.2.1.3.1

Group the first two terms and the last two terms.

$3 ((63 x^{2} - 7 x) - 9 x + 1) = 0$

Step 4.3.2.1.3.2

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

$3 (7 x (9 x - 1) - (9 x - 1)) = 0$

Step 4.3.2.1.4

Factor the polynomial by factoring out the greatest common factor, $9 x - 1$ .

$3 ((9 x - 1) (7 x - 1)) = 0$

Step 4.3.2.2

Remove unnecessary parentheses.

$3 (9 x - 1) (7 x - 1) = 0$

Step 4.4

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

$9 x - 1 = 0$

$7 x - 1 = 0$

Step 4.5

Set $9 x - 1$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $9 x - 1$ equal to $0$ .

$9 x - 1 = 0$

Step 4.5.2

Solve $9 x - 1 = 0$ for $x$ .

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Step 4.5.2.1

Add $1$ to both sides of the equation.

$9 x = 1$

Step 4.5.2.2

Divide each term in $9 x = 1$ by $9$ and simplify.

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Step 4.5.2.2.1

Divide each term in $9 x = 1$ by $9$ .

$\frac{9 x}{9} = \frac{1}{9}$

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Cancel the common factor of $9$ .

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Step 4.5.2.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{1}{9}$

Step 4.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{9}$

Step 4.6

Set $7 x - 1$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $7 x - 1$ equal to $0$ .

$7 x - 1 = 0$

Step 4.6.2

Solve $7 x - 1 = 0$ for $x$ .

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Step 4.6.2.1

Add $1$ to both sides of the equation.

$7 x = 1$

Step 4.6.2.2

Divide each term in $7 x = 1$ by $7$ and simplify.

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Step 4.6.2.2.1

Divide each term in $7 x = 1$ by $7$ .

$\frac{7 x}{7} = \frac{1}{7}$

Step 4.6.2.2.2

Simplify the left side.

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Step 4.6.2.2.2.1

Cancel the common factor of $7$ .

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Step 4.6.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{1}{7}$

Step 4.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{7}$

Step 4.7

The final solution is all the values that make $3 (9 x - 1) (7 x - 1) = 0$ true.

$x = \frac{1}{9}, \frac{1}{7}$

Step 5

Exclude the solutions that do not make $\sqrt{3 x^{2}} - x \sqrt{12} + 2 x \sqrt{75} = \sqrt{3}$ true.

$x = \frac{1}{9}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{9}$

Decimal Form:

$x = 0.\overline{1}$