Basic Math Examples

$- (2 - a - a^{2}) + (3 a^{2} - 5 a + 2) = 120$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$- 1 \cdot 2 - - a - - a^{2} + 3 a^{2} - 5 a + 2 = 120$

Step 1.1.2

Simplify.

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Step 1.1.2.1

Multiply $- 1$ by $2$ .

$- 2 - - a - - a^{2} + 3 a^{2} - 5 a + 2 = 120$

Step 1.1.2.2

Multiply $- - a$ .

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Step 1.1.2.2.1

Multiply $- 1$ by $- 1$ .

$- 2 + 1 a - - a^{2} + 3 a^{2} - 5 a + 2 = 120$

Step 1.1.2.2.2

Multiply $a$ by $1$ .

$- 2 + a - - a^{2} + 3 a^{2} - 5 a + 2 = 120$

Step 1.1.2.3

Multiply $- - a^{2}$ .

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Step 1.1.2.3.1

Multiply $- 1$ by $- 1$ .

$- 2 + a + 1 a^{2} + 3 a^{2} - 5 a + 2 = 120$

Step 1.1.2.3.2

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

$- 2 + a + a^{2} + 3 a^{2} - 5 a + 2 = 120$

Step 1.2

Simplify by adding terms.

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Step 1.2.1

Combine the opposite terms in $- 2 + a + a^{2} + 3 a^{2} - 5 a + 2$ .

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Step 1.2.1.1

Add $- 2$ and $2$ .

$a + a^{2} + 3 a^{2} - 5 a + 0 = 120$

Step 1.2.1.2

Add $a + a^{2} + 3 a^{2} - 5 a$ and $0$ .

$a + a^{2} + 3 a^{2} - 5 a = 120$

Step 1.2.2

Subtract $5 a$ from $a$ .

$a^{2} + 3 a^{2} - 4 a = 120$

Step 1.2.3

Add $a^{2}$ and $3 a^{2}$ .

$4 a^{2} - 4 a = 120$

Step 2

Subtract $120$ from both sides of the equation.

$4 a^{2} - 4 a - 120 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $4$ out of $4 a^{2} - 4 a - 120$ .

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Step 3.1.1

Factor $4$ out of $4 a^{2}$ .

$4 (a^{2}) - 4 a - 120 = 0$

Step 3.1.2

Factor $4$ out of $- 4 a$ .

$4 (a^{2}) + 4 (- a) - 120 = 0$

Step 3.1.3

Factor $4$ out of $- 120$ .

$4 (a^{2}) + 4 (- a) + 4 (- 30) = 0$

Step 3.1.4

Factor $4$ out of $4 (a^{2}) + 4 (- a)$ .

$4 (a^{2} - a) + 4 (- 30) = 0$

Step 3.1.5

Factor $4$ out of $4 (a^{2} - a) + 4 (- 30)$ .

$4 (a^{2} - a - 30) = 0$

Step 3.2

Factor.

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Step 3.2.1

Factor $a^{2} - a - 30$ using the AC method.

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Step 3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 30$ and whose sum is $- 1$ .

$- 6, 5$

Step 3.2.1.2

Write the factored form using these integers.

$4 ((a - 6) (a + 5)) = 0$

Step 3.2.2

Remove unnecessary parentheses.

$4 (a - 6) (a + 5) = 0$

Step 4

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

$a - 6 = 0$

$a + 5 = 0$

Step 5

Set $a - 6$ equal to $0$ and solve for $a$ .

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Step 5.1

Set $a - 6$ equal to $0$ .

$a - 6 = 0$

Step 5.2

Add $6$ to both sides of the equation.

$a = 6$

Step 6

Set $a + 5$ equal to $0$ and solve for $a$ .

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Step 6.1

Set $a + 5$ equal to $0$ .

$a + 5 = 0$

Step 6.2

Subtract $5$ from both sides of the equation.

$a = - 5$

Step 7

The final solution is all the values that make $4 (a - 6) (a + 5) = 0$ true.

$a = 6, - 5$