Basic Math Examples

$2 {(a - 3)}^{2} - 3 {(a + 1)}^{2} + (a - 5) (a - 3) + 4 (a^{2} - 5 a + 1) = 4 a^{2} - 12$

Step 1

Simplify $2 {(a - 3)}^{2} - 3 {(a + 1)}^{2} + (a - 5) (a - 3) + 4 (a^{2} - 5 a + 1)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Expand $(a - 5) (a - 3)$ using the FOIL Method.

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Step 1.1.1.1

Apply the distributive property.

$2 {(a - 3)}^{2} - 3 {(a + 1)}^{2} + a (a - 3) - 5 (a - 3) + 4 (a^{2} - 5 a + 1) = 4 a^{2} - 12$

Step 1.1.1.2

Apply the distributive property.

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

Step 1.1.1.3

Apply the distributive property.

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

Step 1.1.2

Simplify and combine like terms.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Multiply $a$ by $a$ .

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

Multiply $- 5$ by $- 3$ .

$2 {(a - 3)}^{2} - 3 {(a + 1)}^{2} + a^{2} - 3 a - 5 a + 15 + 4 (a^{2} - 5 a + 1) = 4 a^{2} - 12$

Step 1.1.2.2

Subtract $5 a$ from $- 3 a$ .

$2 {(a - 3)}^{2} - 3 {(a + 1)}^{2} + a^{2} - 8 a + 15 + 4 (a^{2} - 5 a + 1) = 4 a^{2} - 12$

Step 1.1.3

Apply the distributive property.

$2 {(a - 3)}^{2} - 3 {(a + 1)}^{2} + a^{2} - 8 a + 15 + 4 a^{2} + 4 (- 5 a) + 4 \cdot 1 = 4 a^{2} - 12$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Multiply $- 5$ by $4$ .

$2 {(a - 3)}^{2} - 3 {(a + 1)}^{2} + a^{2} - 8 a + 15 + 4 a^{2} - 20 a + 4 \cdot 1 = 4 a^{2} - 12$

Step 1.1.4.2

Multiply $4$ by $1$ .

$2 {(a - 3)}^{2} - 3 {(a + 1)}^{2} + a^{2} - 8 a + 15 + 4 a^{2} - 20 a + 4 = 4 a^{2} - 12$

Step 1.2

Simplify by adding terms.

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Step 1.2.1

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

$2 {(a - 3)}^{2} - 3 {(a + 1)}^{2} + 5 a^{2} - 8 a + 15 - 20 a + 4 = 4 a^{2} - 12$

Step 1.2.2

Subtract $20 a$ from $- 8 a$ .

$2 {(a - 3)}^{2} - 3 {(a + 1)}^{2} + 5 a^{2} - 28 a + 15 + 4 = 4 a^{2} - 12$

Step 1.2.3

Add $15$ and $4$ .

$2 {(a - 3)}^{2} - 3 {(a + 1)}^{2} + 5 a^{2} - 28 a + 19 = 4 a^{2} - 12$

Step 2

Move all terms containing $a$ to the left side of the equation.

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Step 2.1

Subtract $4 a^{2}$ from both sides of the equation.

$2 {(a - 3)}^{2} - 3 {(a + 1)}^{2} + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite ${(a - 3)}^{2}$ as $(a - 3) (a - 3)$ .

$2 ((a - 3) (a - 3)) - 3 {(a + 1)}^{2} + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.2

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

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Step 2.2.2.1

Apply the distributive property.

$2 (a (a - 3) - 3 (a - 3)) - 3 {(a + 1)}^{2} + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.2.2

Apply the distributive property.

$2 (a \cdot a + a \cdot - 3 - 3 (a - 3)) - 3 {(a + 1)}^{2} + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.2.3

Apply the distributive property.

$2 (a \cdot a + a \cdot - 3 - 3 a - 3 \cdot - 3) - 3 {(a + 1)}^{2} + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.3

Simplify and combine like terms.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $a$ by $a$ .

$2 (a^{2} + a \cdot - 3 - 3 a - 3 \cdot - 3) - 3 {(a + 1)}^{2} + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.3.1.2

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

$2 (a^{2} - 3 \cdot a - 3 a - 3 \cdot - 3) - 3 {(a + 1)}^{2} + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.3.1.3

Multiply $- 3$ by $- 3$ .

$2 (a^{2} - 3 a - 3 a + 9) - 3 {(a + 1)}^{2} + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.3.2

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

$2 (a^{2} - 6 a + 9) - 3 {(a + 1)}^{2} + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.4

Apply the distributive property.

$2 a^{2} + 2 (- 6 a) + 2 \cdot 9 - 3 {(a + 1)}^{2} + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.5

Simplify.

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Step 2.2.5.1

Multiply $- 6$ by $2$ .

$2 a^{2} - 12 a + 2 \cdot 9 - 3 {(a + 1)}^{2} + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.5.2

Multiply $2$ by $9$ .

$2 a^{2} - 12 a + 18 - 3 {(a + 1)}^{2} + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.6

Rewrite ${(a + 1)}^{2}$ as $(a + 1) (a + 1)$ .

$2 a^{2} - 12 a + 18 - 3 ((a + 1) (a + 1)) + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.7

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

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Step 2.2.7.1

Apply the distributive property.

$2 a^{2} - 12 a + 18 - 3 (a (a + 1) + 1 (a + 1)) + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.7.2

Apply the distributive property.

$2 a^{2} - 12 a + 18 - 3 (a \cdot a + a \cdot 1 + 1 (a + 1)) + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.7.3

Apply the distributive property.

$2 a^{2} - 12 a + 18 - 3 (a \cdot a + a \cdot 1 + 1 a + 1 \cdot 1) + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.8

Simplify and combine like terms.

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Step 2.2.8.1

Simplify each term.

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Step 2.2.8.1.1

Multiply $a$ by $a$ .

$2 a^{2} - 12 a + 18 - 3 (a^{2} + a \cdot 1 + 1 a + 1 \cdot 1) + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.8.1.2

Multiply $a$ by $1$ .

$2 a^{2} - 12 a + 18 - 3 (a^{2} + a + 1 a + 1 \cdot 1) + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.8.1.3

Multiply $a$ by $1$ .

$2 a^{2} - 12 a + 18 - 3 (a^{2} + a + a + 1 \cdot 1) + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.8.1.4

Multiply $1$ by $1$ .

$2 a^{2} - 12 a + 18 - 3 (a^{2} + a + a + 1) + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.8.2

Add $a$ and $a$ .

$2 a^{2} - 12 a + 18 - 3 (a^{2} + 2 a + 1) + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.9

Apply the distributive property.

$2 a^{2} - 12 a + 18 - 3 a^{2} - 3 (2 a) - 3 \cdot 1 + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.10

Simplify.

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Step 2.2.10.1

Multiply $2$ by $- 3$ .

$2 a^{2} - 12 a + 18 - 3 a^{2} - 6 a - 3 \cdot 1 + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.2.10.2

Multiply $- 3$ by $1$ .

$2 a^{2} - 12 a + 18 - 3 a^{2} - 6 a - 3 + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.3

Subtract $3 a^{2}$ from $2 a^{2}$ .

$- a^{2} - 12 a + 18 - 6 a - 3 + 5 a^{2} - 28 a + 19 - 4 a^{2} = - 12$

Step 2.4

Add $- a^{2}$ and $5 a^{2}$ .

$4 a^{2} - 12 a + 18 - 6 a - 3 - 28 a + 19 - 4 a^{2} = - 12$

Step 2.5

Combine the opposite terms in $4 a^{2} - 12 a + 18 - 6 a - 3 - 28 a + 19 - 4 a^{2}$ .

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Step 2.5.1

Subtract $4 a^{2}$ from $4 a^{2}$ .

$- 12 a + 18 - 6 a - 3 - 28 a + 19 + 0 = - 12$

Step 2.5.2

Add $- 12 a + 18 - 6 a - 3 - 28 a + 19$ and $0$ .

$- 12 a + 18 - 6 a - 3 - 28 a + 19 = - 12$

Step 2.6

Subtract $6 a$ from $- 12 a$ .

$- 18 a + 18 - 3 - 28 a + 19 = - 12$

Step 2.7

Subtract $28 a$ from $- 18 a$ .

$- 46 a + 18 - 3 + 19 = - 12$

Step 2.8

Subtract $3$ from $18$ .

$- 46 a + 15 + 19 = - 12$

Step 2.9

Add $15$ and $19$ .

$- 46 a + 34 = - 12$

Step 3

Move all terms not containing $a$ to the right side of the equation.

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Step 3.1

Subtract $34$ from both sides of the equation.

$- 46 a = - 12 - 34$

Step 3.2

Subtract $34$ from $- 12$ .

$- 46 a = - 46$

Step 4

Divide each term in $- 46 a = - 46$ by $- 46$ and simplify.

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Step 4.1

Divide each term in $- 46 a = - 46$ by $- 46$ .

$\frac{- 46 a}{- 46} = \frac{- 46}{- 46}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 46$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 46 a}{- 46} = \frac{- 46}{- 46}$

Step 4.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 46}{- 46}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 46$ by $- 46$ .

$a = 1$