Basic Math Examples

$(a + 2) (2 a - 3) + (a - 5) (a - 4) - (a + 1) (3 a + 2) - \frac{11}{2} = 0$

Step 1

Simplify $(a + 2) (2 a - 3) + (a - 5) (a - 4) - (a + 1) (3 a + 2) - \frac{11}{2}$ .

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Step 1.1

Find the common denominator.

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Step 1.1.1

Write $(a + 2) (2 a - 3)$ as a fraction with denominator $1$ .

$\frac{(a + 2) (2 a - 3)}{1} + (a - 5) (a - 4) - (a + 1) (3 a + 2) - \frac{11}{2} = 0$

Step 1.1.2

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

$\frac{(a + 2) (2 a - 3)}{1} \cdot \frac{2}{2} + (a - 5) (a - 4) - (a + 1) (3 a + 2) - \frac{11}{2} = 0$

Step 1.1.3

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

$\frac{(a + 2) (2 a - 3) \cdot 2}{2} + (a - 5) (a - 4) - (a + 1) (3 a + 2) - \frac{11}{2} = 0$

Step 1.1.4

Write $(a - 5) (a - 4)$ as a fraction with denominator $1$ .

$\frac{(a + 2) (2 a - 3) \cdot 2}{2} + \frac{(a - 5) (a - 4)}{1} - (a + 1) (3 a + 2) - \frac{11}{2} = 0$

Step 1.1.5

Multiply $\frac{(a - 5) (a - 4)}{1}$ by $\frac{2}{2}$ .

$\frac{(a + 2) (2 a - 3) \cdot 2}{2} + \frac{(a - 5) (a - 4)}{1} \cdot \frac{2}{2} - (a + 1) (3 a + 2) - \frac{11}{2} = 0$

Step 1.1.6

Multiply $\frac{(a - 5) (a - 4)}{1}$ by $\frac{2}{2}$ .

$\frac{(a + 2) (2 a - 3) \cdot 2}{2} + \frac{(a - 5) (a - 4) \cdot 2}{2} - (a + 1) (3 a + 2) - \frac{11}{2} = 0$

Step 1.1.7

Write $- (a + 1) (3 a + 2)$ as a fraction with denominator $1$ .

$\frac{(a + 2) (2 a - 3) \cdot 2}{2} + \frac{(a - 5) (a - 4) \cdot 2}{2} + \frac{- (a + 1) (3 a + 2)}{1} - \frac{11}{2} = 0$

Step 1.1.8

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

$\frac{(a + 2) (2 a - 3) \cdot 2}{2} + \frac{(a - 5) (a - 4) \cdot 2}{2} + \frac{- (a + 1) (3 a + 2)}{1} \cdot \frac{2}{2} - \frac{11}{2} = 0$

Step 1.1.9

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

$\frac{(a + 2) (2 a - 3) \cdot 2}{2} + \frac{(a - 5) (a - 4) \cdot 2}{2} + \frac{- (a + 1) (3 a + 2) \cdot 2}{2} - \frac{11}{2} = 0$

Step 1.2

Combine the numerators over the common denominator.

$\frac{(a + 2) (2 a - 3) \cdot 2 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3

Simplify each term.

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Step 1.3.1

Expand $(a + 2) (2 a - 3)$ using the FOIL Method.

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Step 1.3.1.1

Apply the distributive property.

$\frac{(a (2 a - 3) + 2 (2 a - 3)) \cdot 2 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.1.2

Apply the distributive property.

$\frac{(a (2 a) + a \cdot - 3 + 2 (2 a - 3)) \cdot 2 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.1.3

Apply the distributive property.

$\frac{(a (2 a) + a \cdot - 3 + 2 (2 a) + 2 \cdot - 3) \cdot 2 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.2

Simplify and combine like terms.

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Step 1.3.2.1

Simplify each term.

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Step 1.3.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{(2 a \cdot a + a \cdot - 3 + 2 (2 a) + 2 \cdot - 3) \cdot 2 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.2.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 1.3.2.1.2.1

Move $a$ .

$\frac{(2 (a \cdot a) + a \cdot - 3 + 2 (2 a) + 2 \cdot - 3) \cdot 2 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.2.1.2.2

Multiply $a$ by $a$ .

$\frac{(2 a^{2} + a \cdot - 3 + 2 (2 a) + 2 \cdot - 3) \cdot 2 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.2.1.3

Move $- 3$ to the left of $a$ .

$\frac{(2 a^{2} - 3 \cdot a + 2 (2 a) + 2 \cdot - 3) \cdot 2 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.2.1.4

Multiply $2$ by $2$ .

$\frac{(2 a^{2} - 3 a + 4 a + 2 \cdot - 3) \cdot 2 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.2.1.5

Multiply $2$ by $- 3$ .

$\frac{(2 a^{2} - 3 a + 4 a - 6) \cdot 2 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.2.2

Add $- 3 a$ and $4 a$ .

$\frac{(2 a^{2} + a - 6) \cdot 2 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.3

Apply the distributive property.

$\frac{2 a^{2} \cdot 2 + a \cdot 2 - 6 \cdot 2 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.4

Simplify.

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Step 1.3.4.1

Multiply $2$ by $2$ .

$\frac{4 a^{2} + a \cdot 2 - 6 \cdot 2 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.4.2

Move $2$ to the left of $a$ .

$\frac{4 a^{2} + 2 \cdot a - 6 \cdot 2 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.4.3

Multiply $- 6$ by $2$ .

$\frac{4 a^{2} + 2 \cdot a - 12 + (a - 5) (a - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.5

Expand $(a - 5) (a - 4)$ using the FOIL Method.

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Step 1.3.5.1

Apply the distributive property.

$\frac{4 a^{2} + 2 a - 12 + (a (a - 4) - 5 (a - 4)) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.5.2

Apply the distributive property.

$\frac{4 a^{2} + 2 a - 12 + (a \cdot a + a \cdot - 4 - 5 (a - 4)) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.5.3

Apply the distributive property.

$\frac{4 a^{2} + 2 a - 12 + (a \cdot a + a \cdot - 4 - 5 a - 5 \cdot - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.6

Simplify and combine like terms.

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Step 1.3.6.1

Simplify each term.

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Step 1.3.6.1.1

Multiply $a$ by $a$ .

$\frac{4 a^{2} + 2 a - 12 + (a^{2} + a \cdot - 4 - 5 a - 5 \cdot - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.6.1.2

Move $- 4$ to the left of $a$ .

$\frac{4 a^{2} + 2 a - 12 + (a^{2} - 4 \cdot a - 5 a - 5 \cdot - 4) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.6.1.3

Multiply $- 5$ by $- 4$ .

$\frac{4 a^{2} + 2 a - 12 + (a^{2} - 4 a - 5 a + 20) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.6.2

Subtract $5 a$ from $- 4 a$ .

$\frac{4 a^{2} + 2 a - 12 + (a^{2} - 9 a + 20) \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.7

Apply the distributive property.

$\frac{4 a^{2} + 2 a - 12 + a^{2} \cdot 2 - 9 a \cdot 2 + 20 \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.8

Simplify.

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Step 1.3.8.1

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

$\frac{4 a^{2} + 2 a - 12 + 2 \cdot a^{2} - 9 a \cdot 2 + 20 \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.8.2

Multiply $2$ by $- 9$ .

$\frac{4 a^{2} + 2 a - 12 + 2 \cdot a^{2} - 18 a + 20 \cdot 2 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.8.3

Multiply $20$ by $2$ .

$\frac{4 a^{2} + 2 a - 12 + 2 \cdot a^{2} - 18 a + 40 - (a + 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.9

Apply the distributive property.

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 + (- a - 1 \cdot 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.10

Multiply $- 1$ by $1$ .

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 + (- a - 1) (3 a + 2) \cdot 2 - 11}{2} = 0$

Step 1.3.11

Expand $(- a - 1) (3 a + 2)$ using the FOIL Method.

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Step 1.3.11.1

Apply the distributive property.

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 + (- a (3 a + 2) - 1 (3 a + 2)) \cdot 2 - 11}{2} = 0$

Step 1.3.11.2

Apply the distributive property.

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 + (- a (3 a) - a \cdot 2 - 1 (3 a + 2)) \cdot 2 - 11}{2} = 0$

Step 1.3.11.3

Apply the distributive property.

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 + (- a (3 a) - a \cdot 2 - 1 (3 a) - 1 \cdot 2) \cdot 2 - 11}{2} = 0$

Step 1.3.12

Simplify and combine like terms.

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Step 1.3.12.1

Simplify each term.

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Step 1.3.12.1.1

Rewrite using the commutative property of multiplication.

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 + (- 1 \cdot 3 a \cdot a - a \cdot 2 - 1 (3 a) - 1 \cdot 2) \cdot 2 - 11}{2} = 0$

Step 1.3.12.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 1.3.12.1.2.1

Move $a$ .

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 + (- 1 \cdot 3 (a \cdot a) - a \cdot 2 - 1 (3 a) - 1 \cdot 2) \cdot 2 - 11}{2} = 0$

Step 1.3.12.1.2.2

Multiply $a$ by $a$ .

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 + (- 1 \cdot 3 a^{2} - a \cdot 2 - 1 (3 a) - 1 \cdot 2) \cdot 2 - 11}{2} = 0$

Step 1.3.12.1.3

Multiply $- 1$ by $3$ .

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 + (- 3 a^{2} - a \cdot 2 - 1 (3 a) - 1 \cdot 2) \cdot 2 - 11}{2} = 0$

Step 1.3.12.1.4

Multiply $2$ by $- 1$ .

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 + (- 3 a^{2} - 2 a - 1 (3 a) - 1 \cdot 2) \cdot 2 - 11}{2} = 0$

Step 1.3.12.1.5

Multiply $3$ by $- 1$ .

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 + (- 3 a^{2} - 2 a - 3 a - 1 \cdot 2) \cdot 2 - 11}{2} = 0$

Step 1.3.12.1.6

Multiply $- 1$ by $2$ .

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 + (- 3 a^{2} - 2 a - 3 a - 2) \cdot 2 - 11}{2} = 0$

Step 1.3.12.2

Subtract $3 a$ from $- 2 a$ .

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 + (- 3 a^{2} - 5 a - 2) \cdot 2 - 11}{2} = 0$

Step 1.3.13

Apply the distributive property.

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 - 3 a^{2} \cdot 2 - 5 a \cdot 2 - 2 \cdot 2 - 11}{2} = 0$

Step 1.3.14

Simplify.

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Step 1.3.14.1

Multiply $2$ by $- 3$ .

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 - 6 a^{2} - 5 a \cdot 2 - 2 \cdot 2 - 11}{2} = 0$

Step 1.3.14.2

Multiply $2$ by $- 5$ .

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 - 6 a^{2} - 10 a - 2 \cdot 2 - 11}{2} = 0$

Step 1.3.14.3

Multiply $- 2$ by $2$ .

$\frac{4 a^{2} + 2 a - 12 + 2 a^{2} - 18 a + 40 - 6 a^{2} - 10 a - 4 - 11}{2} = 0$

Step 1.4

Simplify terms.

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Step 1.4.1

Add $4 a^{2}$ and $2 a^{2}$ .

$\frac{6 a^{2} + 2 a - 12 - 18 a + 40 - 6 a^{2} - 10 a - 4 - 11}{2} = 0$

Step 1.4.2

Combine the opposite terms in $6 a^{2} + 2 a - 12 - 18 a + 40 - 6 a^{2} - 10 a - 4 - 11$ .

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Step 1.4.2.1

Subtract $6 a^{2}$ from $6 a^{2}$ .

$\frac{2 a - 12 - 18 a + 40 + 0 - 10 a - 4 - 11}{2} = 0$

Step 1.4.2.2

Add $2 a - 12 - 18 a + 40$ and $0$ .

$\frac{2 a - 12 - 18 a + 40 - 10 a - 4 - 11}{2} = 0$

Step 1.4.3

Subtract $18 a$ from $2 a$ .

$\frac{- 16 a - 12 + 40 - 10 a - 4 - 11}{2} = 0$

Step 1.4.4

Subtract $10 a$ from $- 16 a$ .

$\frac{- 26 a - 12 + 40 - 4 - 11}{2} = 0$

Step 1.4.5

Simplify by adding and subtracting.

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Step 1.4.5.1

Add $- 12$ and $40$ .

$\frac{- 26 a + 28 - 4 - 11}{2} = 0$

Step 1.4.5.2

Subtract $4$ from $28$ .

$\frac{- 26 a + 24 - 11}{2} = 0$

Step 1.4.5.3

Subtract $11$ from $24$ .

$\frac{- 26 a + 13}{2} = 0$

Step 1.4.6

Factor $13$ out of $- 26 a + 13$ .

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Step 1.4.6.1

Factor $13$ out of $- 26 a$ .

$\frac{13 (- 2 a) + 13}{2} = 0$

Step 1.4.6.2

Factor $13$ out of $13$ .

$\frac{13 (- 2 a) + 13 (1)}{2} = 0$

Step 1.4.6.3

Factor $13$ out of $13 (- 2 a) + 13 (1)$ .

$\frac{13 (- 2 a + 1)}{2} = 0$

Step 1.4.7

Factor $- 1$ out of $- 2 a$ .

$\frac{13 (- (2 a) + 1)}{2} = 0$

Step 1.4.8

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

$\frac{13 (- (2 a) - 1 (- 1))}{2} = 0$

Step 1.4.9

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

$\frac{13 (- (2 a - 1))}{2} = 0$

Step 1.4.10

Simplify the expression.

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Step 1.4.10.1

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

$\frac{13 (- 1 (2 a - 1))}{2} = 0$

Step 1.4.10.2

Move the negative in front of the fraction.

$- \frac{13 (2 a - 1)}{2} = 0$

Step 2

Set the numerator equal to zero.

$13 (2 a - 1) = 0$

Step 3

Solve the equation for $a$ .

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Step 3.1

Divide each term in $13 (2 a - 1) = 0$ by $13$ and simplify.

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Step 3.1.1

Divide each term in $13 (2 a - 1) = 0$ by $13$ .

$\frac{13 (2 a - 1)}{13} = \frac{0}{13}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $13$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{13 (2 a - 1)}{13} = \frac{0}{13}$

Step 3.1.2.1.2

Divide $2 a - 1$ by $1$ .

$2 a - 1 = \frac{0}{13}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $0$ by $13$ .

$2 a - 1 = 0$

Step 3.2

Add $1$ to both sides of the equation.

$2 a = 1$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $a$ by $1$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$a = 0.5$