Basic Math Examples

$(j^{2} - 2 j) (\frac{- j + 5}{2})$

Step 1

Simplify terms.

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Step 1.1

Apply the distributive property.

$j^{2} \frac{- j + 5}{2} - 2 j \frac{- j + 5}{2}$

Step 1.2

Combine $j^{2}$ and $\frac{- j + 5}{2}$ .

$\frac{j^{2} (- j + 5)}{2} - 2 j \frac{- j + 5}{2}$

Step 1.3

Cancel the common factor of $2$ .

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Step 1.3.1

Factor $2$ out of $- 2 j$ .

$\frac{j^{2} (- j + 5)}{2} + 2 (- j) \frac{- j + 5}{2}$

Step 1.3.2

Cancel the common factor.

$\frac{j^{2} (- j + 5)}{2} + 2 (- j) \frac{- j + 5}{2}$

Step 1.3.3

Rewrite the expression.

$\frac{j^{2} (- j + 5)}{2} - j (- j + 5)$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$\frac{j^{2} (- j + 5)}{2} - j (- j) - j \cdot 5$

Step 2.2

Rewrite using the commutative property of multiplication.

$\frac{j^{2} (- j + 5)}{2} - 1 \cdot - 1 j \cdot j - j \cdot 5$

Step 2.3

Multiply $5$ by $- 1$ .

$\frac{j^{2} (- j + 5)}{2} - 1 \cdot - 1 j \cdot j - 5 j$

Step 2.4

Simplify each term.

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Step 2.4.1

Multiply $j$ by $j$ by adding the exponents.

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Step 2.4.1.1

Move $j$ .

$\frac{j^{2} (- j + 5)}{2} - 1 \cdot - 1 (j \cdot j) - 5 j$

Step 2.4.1.2

Multiply $j$ by $j$ .

$\frac{j^{2} (- j + 5)}{2} - 1 \cdot - 1 j^{2} - 5 j$

Step 2.4.2

Multiply $- 1$ by $- 1$ .

$\frac{j^{2} (- j + 5)}{2} + 1 j^{2} - 5 j$

Step 2.4.3

Multiply $j^{2}$ by $1$ .

$\frac{j^{2} (- j + 5)}{2} + j^{2} - 5 j$

Step 3

To write $j^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{j^{2} (- j + 5)}{2} + j^{2} \cdot \frac{2}{2} - 5 j$

Step 4

Simplify terms.

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Step 4.1

Combine $j^{2}$ and $\frac{2}{2}$ .

$\frac{j^{2} (- j + 5)}{2} + \frac{j^{2} \cdot 2}{2} - 5 j$

Step 4.2

Combine the numerators over the common denominator.

$\frac{j^{2} (- j + 5) + j^{2} \cdot 2}{2} - 5 j$

Step 5

Simplify the numerator.

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Step 5.1

Factor $j^{2}$ out of $j^{2} (- j + 5) + j^{2} \cdot 2$ .

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Step 5.1.1

Factor $j^{2}$ out of $j^{2} \cdot 2$ .

$\frac{j^{2} (- j + 5) + j^{2} (2)}{2} - 5 j$

Step 5.1.2

Factor $j^{2}$ out of $j^{2} (- j + 5) + j^{2} (2)$ .

$\frac{j^{2} (- j + 5 + 2)}{2} - 5 j$

Step 5.2

Add $5$ and $2$ .

$\frac{j^{2} (- j + 7)}{2} - 5 j$

Step 6

To write $- 5 j$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{j^{2} (- j + 7)}{2} - 5 j \cdot \frac{2}{2}$

Step 7

Simplify terms.

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Step 7.1

Combine $- 5 j$ and $\frac{2}{2}$ .

$\frac{j^{2} (- j + 7)}{2} + \frac{- 5 j \cdot 2}{2}$

Step 7.2

Combine the numerators over the common denominator.

$\frac{j^{2} (- j + 7) - 5 j \cdot 2}{2}$

Step 8

Simplify the numerator.

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Step 8.1

Factor $j$ out of $j^{2} (- j + 7) - 5 j \cdot 2$ .

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Step 8.1.1

Factor $j$ out of $j^{2} (- j + 7)$ .

$\frac{j (j (- j + 7)) - 5 j \cdot 2}{2}$

Step 8.1.2

Factor $j$ out of $- 5 j \cdot 2$ .

$\frac{j (j (- j + 7)) + j (- 5 \cdot 2)}{2}$

Step 8.1.3

Factor $j$ out of $j (j (- j + 7)) + j (- 5 \cdot 2)$ .

$\frac{j (j (- j + 7) - 5 \cdot 2)}{2}$

Step 8.2

Apply the distributive property.

$\frac{j (j (- j) + j \cdot 7 - 5 \cdot 2)}{2}$

Step 8.3

Rewrite using the commutative property of multiplication.

$\frac{j (- j \cdot j + j \cdot 7 - 5 \cdot 2)}{2}$

Step 8.4

Move $7$ to the left of $j$ .

$\frac{j (- j \cdot j + 7 \cdot j - 5 \cdot 2)}{2}$

Step 8.5

Multiply $j$ by $j$ by adding the exponents.

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Step 8.5.1

Move $j$ .

$\frac{j (- (j \cdot j) + 7 \cdot j - 5 \cdot 2)}{2}$

Step 8.5.2

Multiply $j$ by $j$ .

$\frac{j (- j^{2} + 7 \cdot j - 5 \cdot 2)}{2}$

$\frac{j (- j^{2} + 7 j - 5 \cdot 2)}{2}$

Step 8.6

Multiply $- 5$ by $2$ .

$\frac{j (- j^{2} + 7 j - 10)}{2}$

Step 8.7

Factor by grouping.

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Step 8.7.1

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 $a \cdot c = - 1 \cdot - 10 = 10$ and whose sum is $b = 7$ .

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Step 8.7.1.1

Factor $7$ out of $7 j$ .

$\frac{j (- j^{2} + 7 (j) - 10)}{2}$

Step 8.7.1.2

Rewrite $7$ as $2$ plus $5$

$\frac{j (- j^{2} + (2 + 5) j - 10)}{2}$

Step 8.7.1.3

Apply the distributive property.

$\frac{j (- j^{2} + 2 j + 5 j - 10)}{2}$

Step 8.7.2

Factor out the greatest common factor from each group.

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Step 8.7.2.1

Group the first two terms and the last two terms.

$\frac{j ((- j^{2} + 2 j) + 5 j - 10)}{2}$

Step 8.7.2.2

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

$\frac{j (j (- j + 2) - 5 (- j + 2))}{2}$

Step 8.7.3

Factor the polynomial by factoring out the greatest common factor, $- j + 2$ .

$\frac{j ((- j + 2) (j - 5))}{2}$

$\frac{j (- j + 2) (j - 5)}{2}$

Step 9

Simplify with factoring out.

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Step 9.1

Factor $- 1$ out of $- j$ .

$\frac{j (- (j) + 2) (j - 5)}{2}$

Step 9.2

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{j (- (j) - 1 (- 2)) (j - 5)}{2}$

Step 9.3

Factor $- 1$ out of $- (j) - 1 (- 2)$ .

$\frac{j (- (j - 2)) (j - 5)}{2}$

Step 9.4

Simplify the expression.

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Step 9.4.1

Rewrite $- (j - 2)$ as $- 1 (j - 2)$ .

$\frac{j (- 1 (j - 2)) (j - 5)}{2}$

Step 9.4.2

Move the negative in front of the fraction.

$- \frac{(j (j - 2)) (j - 5)}{2}$

Step 9.4.3

Reorder factors in $- \frac{(j (j - 2)) (j - 5)}{2}$ .

$- \frac{j (j - 2) (j - 5)}{2}$