Algebra Examples

$3 x = \frac{5 x + 6}{2 x + 3}$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, 2 x + 3$

Step 1.2

Remove parentheses.

$1, 2 x + 3$

Step 1.3

The LCM of one and any expression is the expression.

$2 x + 3$

Step 2

Multiply each term in $3 x = \frac{5 x + 6}{2 x + 3}$ by $2 x + 3$ to eliminate the fractions.

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Step 2.1

Multiply each term in $3 x = \frac{5 x + 6}{2 x + 3}$ by $2 x + 3$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify by multiplying through.

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Simplify the expression.

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Step 2.2.1.2.1

Rewrite using the commutative property of multiplication.

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

Step 2.2.1.2.2

Multiply $3$ by $3$ .

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

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

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

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Step 2.2.2.1.1

Move $x$ .

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

Step 2.2.2.1.2

Multiply $x$ by $x$ .

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

Step 2.2.2.2

Multiply $3$ by $2$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $2 x + 3$ .

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Step 2.3.1.1

Cancel the common factor.

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

Step 2.3.1.2

Rewrite the expression.

$6 x^{2} + 9 x = 5 x + 6$

Step 3

Solve the equation.

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Step 3.1

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

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Step 3.1.1

Subtract $5 x$ from both sides of the equation.

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

Step 3.1.2

Subtract $5 x$ from $9 x$ .

$6 x^{2} + 4 x = 6$

Step 3.2

Subtract $6$ from both sides of the equation.

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

Step 3.3

Factor $2$ out of $6 x^{2} + 4 x - 6$ .

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Step 3.3.1

Factor $2$ out of $6 x^{2}$ .

$2 (3 x^{2}) + 4 x - 6 = 0$

Step 3.3.2

Factor $2$ out of $4 x$ .

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

Step 3.3.3

Factor $2$ out of $- 6$ .

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.4

Divide each term in $2 (3 x^{2} + 2 x - 3) = 0$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 (3 x^{2} + 2 x - 3) = 0$ by $2$ .

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

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

Step 3.4.2.1.2

Divide $3 x^{2} + 2 x - 3$ by $1$ .

$3 x^{2} + 2 x - 3 = \frac{0}{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $0$ by $2$ .

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

Step 3.5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.6

Substitute the values $a = 3$ , $b = 2$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (3 \cdot - 3)}}{2 \cdot 3}$

Step 3.7

Simplify.

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 3 \cdot - 3}}{2 \cdot 3}$

Step 3.7.1.2

Multiply $- 4 \cdot 3 \cdot - 3$ .

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Step 3.7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 2 \pm \sqrt{4 - 12 \cdot - 3}}{2 \cdot 3}$

Step 3.7.1.2.2

Multiply $- 12$ by $- 3$ .

$x = \frac{- 2 \pm \sqrt{4 + 36}}{2 \cdot 3}$

Step 3.7.1.3

Add $4$ and $36$ .

$x = \frac{- 2 \pm \sqrt{40}}{2 \cdot 3}$

Step 3.7.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 3.7.1.4.1

Factor $4$ out of $40$ .

$x = \frac{- 2 \pm \sqrt{4 (10)}}{2 \cdot 3}$

Step 3.7.1.4.2

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 3}$

Step 3.7.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{10}}{2 \cdot 3}$

Step 3.7.2

Multiply $2$ by $3$ .

$x = \frac{- 2 \pm 2 \sqrt{10}}{6}$

Step 3.7.3

Simplify $\frac{- 2 \pm 2 \sqrt{10}}{6}$ .

$x = \frac{- 1 \pm \sqrt{10}}{3}$

Step 3.8

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{10}}{3}, - \frac{1 + \sqrt{10}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{10}}{3}, - \frac{1 + \sqrt{10}}{3}$

Decimal Form:

$x = 0.72075922 \dots, - 1.38742588 \dots$