Algebra Examples

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

Step 1

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

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

Step 2

Subtract $1$ from both sides of the equation.

$2 \sqrt{x} = 3 x - 1$

Step 3

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

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

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

Apply the product rule to $2 x^{\frac{1}{2}}$ .

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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Step 4.2.1.3.1

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

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

Step 4.2.1.3.2

Cancel the common factor of $2$ .

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Step 4.2.1.3.2.1

Cancel the common factor.

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

Step 4.2.1.3.2.2

Rewrite the expression.

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

Step 4.2.1.4

Simplify.

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

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

Rewrite ${(3 x - 1)}^{2}$ as $(3 x - 1) (3 x - 1)$ .

$4 x = (3 x - 1) (3 x - 1)$

Step 4.3.1.2

Expand $(3 x - 1) (3 x - 1)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$4 x = 3 x (3 x - 1) - 1 (3 x - 1)$

Step 4.3.1.2.2

Apply the distributive property.

$4 x = 3 x (3 x) + 3 x \cdot - 1 - 1 (3 x - 1)$

Step 4.3.1.2.3

Apply the distributive property.

$4 x = 3 x (3 x) + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1$

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

Rewrite using the commutative property of multiplication.

$4 x = 3 \cdot 3 x \cdot x + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1$

Step 4.3.1.3.1.2

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

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Step 4.3.1.3.1.2.1

Move $x$ .

$4 x = 3 \cdot 3 (x \cdot x) + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1$

Step 4.3.1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 4.3.1.3.1.3

Multiply $3$ by $3$ .

$4 x = 9 x^{2} + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1$

Step 4.3.1.3.1.4

Multiply $- 1$ by $3$ .

$4 x = 9 x^{2} - 3 x - 1 (3 x) - 1 \cdot - 1$

Step 4.3.1.3.1.5

Multiply $3$ by $- 1$ .

$4 x = 9 x^{2} - 3 x - 3 x - 1 \cdot - 1$

Step 4.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$4 x = 9 x^{2} - 3 x - 3 x + 1$

Step 4.3.1.3.2

Subtract $3 x$ from $- 3 x$ .

$4 x = 9 x^{2} - 6 x + 1$

Step 5

Solve for $x$ .

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Step 5.1

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

$9 x^{2} - 6 x + 1 = 4 x$

Step 5.2

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

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Step 5.2.1

Subtract $4 x$ from both sides of the equation.

$9 x^{2} - 6 x + 1 - 4 x = 0$

Step 5.2.2

Subtract $4 x$ from $- 6 x$ .

$9 x^{2} - 10 x + 1 = 0$

Step 5.3

Factor by grouping.

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Step 5.3.1

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 $a \cdot c = 9 \cdot 1 = 9$ and whose sum is $b = - 10$ .

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Step 5.3.1.1

Factor $- 10$ out of $- 10 x$ .

$9 x^{2} - 10 x + 1 = 0$

Step 5.3.1.2

Rewrite $- 10$ as $- 1$ plus $- 9$

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

Step 5.3.1.3

Apply the distributive property.

$9 x^{2} - 1 x - 9 x + 1 = 0$

Step 5.3.2

Factor out the greatest common factor from each group.

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Step 5.3.2.1

Group the first two terms and the last two terms.

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

Step 5.3.2.2

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

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

Step 5.3.3

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

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

Step 5.4

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

$9 x - 1 = 0$

$x - 1 = 0$

Step 5.5

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

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Step 5.5.1

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

$9 x - 1 = 0$

Step 5.5.2

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

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Step 5.5.2.1

Add $1$ to both sides of the equation.

$9 x = 1$

Step 5.5.2.2

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

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Step 5.5.2.2.1

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

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

Step 5.5.2.2.2

Simplify the left side.

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Step 5.5.2.2.2.1

Cancel the common factor of $9$ .

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Step 5.5.2.2.2.1.1

Cancel the common factor.

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

Step 5.5.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 5.6

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

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Step 5.6.1

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

$x - 1 = 0$

Step 5.6.2

Add $1$ to both sides of the equation.

$x = 1$

Step 5.7

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

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

Step 6

Exclude the solutions that do not make $3 x = 1 + 2 \sqrt{x}$ true.

$x = 1$