Basic Math Examples

$3 d - \frac{d + 1}{4 d}$

Step 1

To write $3 d$ as a fraction with a common denominator, multiply by $\frac{4 d}{4 d}$ .

$3 d \cdot \frac{4 d}{4 d} - \frac{d + 1}{4 d}$

Step 2

Simplify terms.

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Step 2.1

Combine $3 d$ and $\frac{4 d}{4 d}$ .

$\frac{3 d (4 d)}{4 d} - \frac{d + 1}{4 d}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{3 d (4 d) - (d + 1)}{4 d}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite using the commutative property of multiplication.

$\frac{3 \cdot 4 d \cdot d - (d + 1)}{4 d}$

Step 3.2

Multiply $d$ by $d$ by adding the exponents.

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Step 3.2.1

Move $d$ .

$\frac{3 \cdot 4 (d \cdot d) - (d + 1)}{4 d}$

Step 3.2.2

Multiply $d$ by $d$ .

$\frac{3 \cdot 4 d^{2} - (d + 1)}{4 d}$

Step 3.3

Multiply $3$ by $4$ .

$\frac{12 d^{2} - (d + 1)}{4 d}$

Step 3.4

Apply the distributive property.

$\frac{12 d^{2} - d - 1 \cdot 1}{4 d}$

Step 3.5

Multiply $- 1$ by $1$ .

$\frac{12 d^{2} - d - 1}{4 d}$

Step 3.6

Factor by grouping.

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Step 3.6.1

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 $a \cdot c = 12 \cdot - 1 = - 12$ and whose sum is $b = - 1$ .

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Step 3.6.1.1

Factor $- 1$ out of $- d$ .

$\frac{12 d^{2} - (d) - 1}{4 d}$

Step 3.6.1.2

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

$\frac{12 d^{2} + (3 - 4) d - 1}{4 d}$

Step 3.6.1.3

Apply the distributive property.

$\frac{12 d^{2} + 3 d - 4 d - 1}{4 d}$

Step 3.6.2

Factor out the greatest common factor from each group.

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Step 3.6.2.1

Group the first two terms and the last two terms.

$\frac{(12 d^{2} + 3 d) - 4 d - 1}{4 d}$

Step 3.6.2.2

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

$\frac{3 d (4 d + 1) - (4 d + 1)}{4 d}$

Step 3.6.3

Factor the polynomial by factoring out the greatest common factor, $4 d + 1$ .

$\frac{(4 d + 1) (3 d - 1)}{4 d}$