Algebra Examples

$\frac{d}{3} + \frac{1}{2} = \frac{1}{3 d}$

Step 1

Subtract $\frac{1}{3 d}$ from both sides of the equation.

$\frac{d}{3} + \frac{1}{2} - \frac{1}{3 d} = 0$

Step 2

Reorder terms.

$\frac{d}{3} - \frac{1}{3 d} + \frac{1}{2} = 0$

Step 3

To write $\frac{d}{3}$ as a fraction with a common denominator, multiply by $\frac{d}{d}$ .

$\frac{d}{3} \cdot \frac{d}{d} - \frac{1}{3 d} + \frac{1}{2} = 0$

Step 4

Multiply $\frac{d}{3}$ by $\frac{d}{d}$ .

$\frac{d \cdot d}{3 d} - \frac{1}{3 d} + \frac{1}{2} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{d \cdot d - 1}{3 d} + \frac{1}{2} = 0$

Step 6

Simplify the numerator.

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Step 6.1

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

$\frac{d^{1 + 1} - 1}{3 d} + \frac{1}{2} = 0$

Step 6.2

Add $1$ and $1$ .

$\frac{d^{2} - 1}{3 d} + \frac{1}{2} = 0$

Step 6.3

Rewrite $1$ as $1^{2}$ .

$\frac{d^{2} - 1^{2}}{3 d} + \frac{1}{2} = 0$

Step 6.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = d$ and $b = 1$ .

$\frac{(d + 1) (d - 1)}{3 d} + \frac{1}{2} = 0$

Step 7

To write $\frac{(d + 1) (d - 1)}{3 d}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{(d + 1) (d - 1)}{3 d} \cdot \frac{2}{2} + \frac{1}{2} = 0$

Step 8

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3 d}{3 d}$ .

$\frac{(d + 1) (d - 1)}{3 d} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{3 d}{3 d} = 0$

Step 9

Write each expression with a common denominator of $6 d$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

Multiply $\frac{(d + 1) (d - 1)}{3 d}$ by $\frac{2}{2}$ .

$\frac{(d + 1) (d - 1) \cdot 2}{3 d \cdot 2} + \frac{1}{2} \cdot \frac{3 d}{3 d} = 0$

Step 9.2

Multiply $2$ by $3$ .

$\frac{(d + 1) (d - 1) \cdot 2}{6 d} + \frac{1}{2} \cdot \frac{3 d}{3 d} = 0$

Step 9.3

Multiply $\frac{1}{2}$ by $\frac{3 d}{3 d}$ .

$\frac{(d + 1) (d - 1) \cdot 2}{6 d} + \frac{3 d}{2 (3 d)} = 0$

Step 9.4

Multiply $3$ by $2$ .

$\frac{(d + 1) (d - 1) \cdot 2}{6 d} + \frac{3 d}{6 d} = 0$

Step 10

Combine the numerators over the common denominator.

$\frac{(d + 1) (d - 1) \cdot 2 + 3 d}{6 d} = 0$

Step 11

Simplify the numerator.

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Step 11.1

Expand $(d + 1) (d - 1)$ using the FOIL Method.

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Step 11.1.1

Apply the distributive property.

$\frac{(d (d - 1) + 1 (d - 1)) \cdot 2 + 3 d}{6 d} = 0$

Step 11.1.2

Apply the distributive property.

$\frac{(d \cdot d + d \cdot - 1 + 1 (d - 1)) \cdot 2 + 3 d}{6 d} = 0$

Step 11.1.3

Apply the distributive property.

$\frac{(d \cdot d + d \cdot - 1 + 1 d + 1 \cdot - 1) \cdot 2 + 3 d}{6 d} = 0$

Step 11.2

Simplify and combine like terms.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Multiply $d$ by $d$ .

$\frac{(d^{2} + d \cdot - 1 + 1 d + 1 \cdot - 1) \cdot 2 + 3 d}{6 d} = 0$

Step 11.2.1.2

Move $- 1$ to the left of $d$ .

$\frac{(d^{2} - 1 d + 1 d + 1 \cdot - 1) \cdot 2 + 3 d}{6 d} = 0$

Step 11.2.1.3

Rewrite $- 1 d$ as $- d$ .

$\frac{(d^{2} - d + 1 d + 1 \cdot - 1) \cdot 2 + 3 d}{6 d} = 0$

Step 11.2.1.4

Multiply $d$ by $1$ .

$\frac{(d^{2} - d + d + 1 \cdot - 1) \cdot 2 + 3 d}{6 d} = 0$

Step 11.2.1.5

Multiply $- 1$ by $1$ .

$\frac{(d^{2} - d + d - 1) \cdot 2 + 3 d}{6 d} = 0$

Step 11.2.2

Add $- d$ and $d$ .

$\frac{(d^{2} + 0 - 1) \cdot 2 + 3 d}{6 d} = 0$

Step 11.2.3

Add $d^{2}$ and $0$ .

$\frac{(d^{2} - 1) \cdot 2 + 3 d}{6 d} = 0$

Step 11.3

Apply the distributive property.

$\frac{d^{2} \cdot 2 - 1 \cdot 2 + 3 d}{6 d} = 0$

Step 11.4

Move $2$ to the left of $d^{2}$ .

$\frac{2 \cdot d^{2} - 1 \cdot 2 + 3 d}{6 d} = 0$

Step 11.5

Multiply $- 1$ by $2$ .

$\frac{2 d^{2} - 2 + 3 d}{6 d} = 0$

Step 11.6

Reorder terms.

$\frac{2 d^{2} + 3 d - 2}{6 d} = 0$

Step 11.7

Factor by grouping.

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Step 11.7.1

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 $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = 3$ .

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Step 11.7.1.1

Factor $3$ out of $3 d$ .

$\frac{2 d^{2} + 3 (d) - 2}{6 d} = 0$

Step 11.7.1.2

Rewrite $3$ as $- 1$ plus $4$

$\frac{2 d^{2} + (- 1 + 4) d - 2}{6 d} = 0$

Step 11.7.1.3

Apply the distributive property.

$\frac{2 d^{2} - 1 d + 4 d - 2}{6 d} = 0$

Step 11.7.2

Factor out the greatest common factor from each group.

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Step 11.7.2.1

Group the first two terms and the last two terms.

$\frac{(2 d^{2} - 1 d) + 4 d - 2}{6 d} = 0$

Step 11.7.2.2

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

$\frac{d (2 d - 1) + 2 (2 d - 1)}{6 d} = 0$

Step 11.7.3

Factor the polynomial by factoring out the greatest common factor, $2 d - 1$ .

$\frac{(2 d - 1) (d + 2)}{6 d} = 0$

Step 12

Set the numerator equal to zero.

$(2 d - 1) (d + 2) = 0$

Step 13

Solve the equation for $d$ .

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Step 13.1

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

$2 d - 1 = 0$

$d + 2 = 0$

Step 13.2

Set $2 d - 1$ equal to $0$ and solve for $d$ .

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Step 13.2.1

Set $2 d - 1$ equal to $0$ .

$2 d - 1 = 0$

Step 13.2.2

Solve $2 d - 1 = 0$ for $d$ .

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Step 13.2.2.1

Add $1$ to both sides of the equation.

$2 d = 1$

Step 13.2.2.2

Divide each term in $2 d = 1$ by $2$ and simplify.

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Step 13.2.2.2.1

Divide each term in $2 d = 1$ by $2$ .

$\frac{2 d}{2} = \frac{1}{2}$

Step 13.2.2.2.2

Simplify the left side.

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Step 13.2.2.2.2.1

Cancel the common factor of $2$ .

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Step 13.2.2.2.2.1.1

Cancel the common factor.

$\frac{2 d}{2} = \frac{1}{2}$

Step 13.2.2.2.2.1.2

Divide $d$ by $1$ .

$d = \frac{1}{2}$

Step 13.3

Set $d + 2$ equal to $0$ and solve for $d$ .

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Step 13.3.1

Set $d + 2$ equal to $0$ .

$d + 2 = 0$

Step 13.3.2

Subtract $2$ from both sides of the equation.

$d = - 2$

Step 13.4

The final solution is all the values that make $(2 d - 1) (d + 2) = 0$ true.

$d = \frac{1}{2}, - 2$