Algebra Examples

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

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.

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

Step 2

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

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

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(3 x - 5)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(3 x - 5)}^{2}$ as $(3 x - 5) (3 x - 5)$ .

$6 x + 5 = (3 x - 5) (3 x - 5)$

Step 3.3.1.2

Expand $(3 x - 5) (3 x - 5)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$6 x + 5 = 3 x (3 x - 5) - 5 (3 x - 5)$

Step 3.3.1.2.2

Apply the distributive property.

$6 x + 5 = 3 x (3 x) + 3 x \cdot - 5 - 5 (3 x - 5)$

Step 3.3.1.2.3

Apply the distributive property.

$6 x + 5 = 3 x (3 x) + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5$

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

Rewrite using the commutative property of multiplication.

$6 x + 5 = 3 \cdot 3 x \cdot x + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5$

Step 3.3.1.3.1.2

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

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Step 3.3.1.3.1.2.1

Move $x$ .

$6 x + 5 = 3 \cdot 3 (x \cdot x) + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5$

Step 3.3.1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.3.1.3.1.3

Multiply $3$ by $3$ .

$6 x + 5 = 9 x^{2} + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5$

Step 3.3.1.3.1.4

Multiply $- 5$ by $3$ .

$6 x + 5 = 9 x^{2} - 15 x - 5 (3 x) - 5 \cdot - 5$

Step 3.3.1.3.1.5

Multiply $3$ by $- 5$ .

$6 x + 5 = 9 x^{2} - 15 x - 15 x - 5 \cdot - 5$

Step 3.3.1.3.1.6

Multiply $- 5$ by $- 5$ .

$6 x + 5 = 9 x^{2} - 15 x - 15 x + 25$

Step 3.3.1.3.2

Subtract $15 x$ from $- 15 x$ .

$6 x + 5 = 9 x^{2} - 30 x + 25$

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.

$9 x^{2} - 30 x + 25 = 6 x + 5$

Step 4.2

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

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Step 4.2.1

Subtract $6 x$ from both sides of the equation.

$9 x^{2} - 30 x + 25 - 6 x = 5$

Step 4.2.2

Subtract $6 x$ from $- 30 x$ .

$9 x^{2} - 36 x + 25 = 5$

Step 4.3

Subtract $5$ from both sides of the equation.

$9 x^{2} - 36 x + 25 - 5 = 0$

Step 4.4

Subtract $5$ from $25$ .

$9 x^{2} - 36 x + 20 = 0$

Step 4.5

Factor by grouping.

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Step 4.5.1

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 $a \cdot c = 9 \cdot 20 = 180$ and whose sum is $b = - 36$ .

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Step 4.5.1.1

Factor $- 36$ out of $- 36 x$ .

$9 x^{2} - 36 x + 20 = 0$

Step 4.5.1.2

Rewrite $- 36$ as $- 6$ plus $- 30$

$9 x^{2} + (- 6 - 30) x + 20 = 0$

Step 4.5.1.3

Apply the distributive property.

$9 x^{2} - 6 x - 30 x + 20 = 0$

Step 4.5.2

Factor out the greatest common factor from each group.

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Step 4.5.2.1

Group the first two terms and the last two terms.

$(9 x^{2} - 6 x) - 30 x + 20 = 0$

Step 4.5.2.2

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

$3 x (3 x - 2) - 10 (3 x - 2) = 0$

Step 4.5.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

$(3 x - 2) (3 x - 10) = 0$

Step 4.6

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

$3 x - 2 = 0$

$3 x - 10 = 0$

Step 4.7

Set $3 x - 2$ equal to $0$ and solve for $x$ .

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Step 4.7.1

Set $3 x - 2$ equal to $0$ .

$3 x - 2 = 0$

Step 4.7.2

Solve $3 x - 2 = 0$ for $x$ .

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Step 4.7.2.1

Add $2$ to both sides of the equation.

$3 x = 2$

Step 4.7.2.2

Divide each term in $3 x = 2$ by $3$ and simplify.

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Step 4.7.2.2.1

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Step 4.7.2.2.2

Simplify the left side.

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Step 4.7.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.7.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Step 4.7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Step 4.8

Set $3 x - 10$ equal to $0$ and solve for $x$ .

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Step 4.8.1

Set $3 x - 10$ equal to $0$ .

$3 x - 10 = 0$

Step 4.8.2

Solve $3 x - 10 = 0$ for $x$ .

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Step 4.8.2.1

Add $10$ to both sides of the equation.

$3 x = 10$

Step 4.8.2.2

Divide each term in $3 x = 10$ by $3$ and simplify.

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Step 4.8.2.2.1

Divide each term in $3 x = 10$ by $3$ .

$\frac{3 x}{3} = \frac{10}{3}$

Step 4.8.2.2.2

Simplify the left side.

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Step 4.8.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.8.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{10}{3}$

Step 4.8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{3}$

Step 4.9

The final solution is all the values that make $(3 x - 2) (3 x - 10) = 0$ true.

$x = \frac{2}{3}, \frac{10}{3}$

Step 5

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

$x = \frac{10}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{10}{3}$

Decimal Form:

$x = 3.\overline{3}$

Mixed Number Form:

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