Algebra Examples

$\frac{10}{x} + 1 = 3 x$

Step 1

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

$\frac{10}{x} + 1 - 3 x = 0$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$x, 1, 1, 1$

Step 2.2

The LCM of one and any expression is the expression.

$x$

Step 3

Multiply each term in $\frac{10}{x} + 1 - 3 x = 0$ by $x$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{10}{x} + 1 - 3 x = 0$ by $x$ .

$\frac{10}{x} x + 1 x - 3 x \cdot x = 0 x$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Cancel the common factor of $x$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{10}{x} x + 1 x - 3 x \cdot x = 0 x$

Step 3.2.1.1.2

Rewrite the expression.

$10 + 1 x - 3 x \cdot x = 0 x$

Step 3.2.1.2

Multiply $x$ by $1$ .

$10 + x - 3 x \cdot x = 0 x$

Step 3.2.1.3

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

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Step 3.2.1.3.1

Move $x$ .

$10 + x - 3 (x \cdot x) = 0 x$

Step 3.2.1.3.2

Multiply $x$ by $x$ .

$10 + x - 3 x^{2} = 0 x$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $0$ by $x$ .

$10 + x - 3 x^{2} = 0$

Step 4

Solve the equation.

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Step 4.1

Factor the left side of the equation.

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Step 4.1.1

Factor $- 1$ out of $10 + x - 3 x^{2}$ .

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Step 4.1.1.1

Reorder the expression.

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Step 4.1.1.1.1

Move $10$ .

$x - 3 x^{2} + 10 = 0$

Step 4.1.1.1.2

Reorder $x$ and $- 3 x^{2}$ .

$- 3 x^{2} + x + 10 = 0$

Step 4.1.1.2

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

$- (3 x^{2}) + x + 10 = 0$

Step 4.1.1.3

Factor $- 1$ out of $x$ .

$- (3 x^{2}) - 1 (- x) + 10 = 0$

Step 4.1.1.4

Rewrite $10$ as $- 1 (- 10)$ .

$- (3 x^{2}) - 1 (- x) - 1 \cdot - 10 = 0$

Step 4.1.1.5

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

$- (3 x^{2} - x) - 1 \cdot - 10 = 0$

Step 4.1.1.6

Factor $- 1$ out of $- (3 x^{2} - x) - 1 (- 10)$ .

$- (3 x^{2} - x - 10) = 0$

Step 4.1.2

Factor.

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Step 4.1.2.1

Factor by grouping.

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Step 4.1.2.1.1

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 $a \cdot c = 3 \cdot - 10 = - 30$ and whose sum is $b = - 1$ .

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Step 4.1.2.1.1.1

Factor $- 1$ out of $- x$ .

$- (3 x^{2} - (x) - 10) = 0$

Step 4.1.2.1.1.2

Rewrite $- 1$ as $5$ plus $- 6$

$- (3 x^{2} + (5 - 6) x - 10) = 0$

Step 4.1.2.1.1.3

Apply the distributive property.

$- (3 x^{2} + 5 x - 6 x - 10) = 0$

Step 4.1.2.1.2

Factor out the greatest common factor from each group.

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Step 4.1.2.1.2.1

Group the first two terms and the last two terms.

$- ((3 x^{2} + 5 x) - 6 x - 10) = 0$

Step 4.1.2.1.2.2

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

$- (x (3 x + 5) - 2 (3 x + 5)) = 0$

Step 4.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 5$ .

$- ((3 x + 5) (x - 2)) = 0$

Step 4.1.2.2

Remove unnecessary parentheses.

$- (3 x + 5) (x - 2) = 0$

Step 4.2

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

$3 x + 5 = 0$

$x - 2 = 0$

Step 4.3

Set $3 x + 5$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $3 x + 5$ equal to $0$ .

$3 x + 5 = 0$

Step 4.3.2

Solve $3 x + 5 = 0$ for $x$ .

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Step 4.3.2.1

Subtract $5$ from both sides of the equation.

$3 x = - 5$

Step 4.3.2.2

Divide each term in $3 x = - 5$ by $3$ and simplify.

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Step 4.3.2.2.1

Divide each term in $3 x = - 5$ by $3$ .

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 4.3.2.2.2

Simplify the left side.

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Step 4.3.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 4.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{3}$

Step 4.3.2.2.3

Simplify the right side.

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Step 4.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{3}$

Step 4.4

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

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Step 4.4.1

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

$x - 2 = 0$

Step 4.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4.5

The final solution is all the values that make $- (3 x + 5) (x - 2) = 0$ true.

$x = - \frac{5}{3}, 2$