Basic Math Examples

${(2 a - 3)}^{2} + (a - 1) (a + 2) = 2 - 11 a$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Apply the distributive property.

${(2 a - 3)}^{2} + a (a + 2) - 1 (a + 2) = 2 - 11 a$

Step 1.1.2

Apply the distributive property.

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

Step 1.1.3

Apply the distributive property.

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

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $a$ by $a$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Rewrite $- 1 a$ as $- a$ .

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

Step 1.2.1.4

Multiply $- 1$ by $2$ .

${(2 a - 3)}^{2} + a^{2} + 2 a - a - 2 = 2 - 11 a$

Step 1.2.2

Subtract $a$ from $2 a$ .

${(2 a - 3)}^{2} + a^{2} + a - 2 = 2 - 11 a$

Step 2

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

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Step 2.1

Add $11 a$ to both sides of the equation.

${(2 a - 3)}^{2} + a^{2} + a - 2 + 11 a = 2$

Step 2.2

Simplify each term.

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Step 2.2.1

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

$(2 a - 3) (2 a - 3) + a^{2} + a - 2 + 11 a = 2$

Step 2.2.2

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

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Step 2.2.2.1

Apply the distributive property.

$2 a (2 a - 3) - 3 (2 a - 3) + a^{2} + a - 2 + 11 a = 2$

Step 2.2.2.2

Apply the distributive property.

$2 a (2 a) + 2 a \cdot - 3 - 3 (2 a - 3) + a^{2} + a - 2 + 11 a = 2$

Step 2.2.2.3

Apply the distributive property.

$2 a (2 a) + 2 a \cdot - 3 - 3 (2 a) - 3 \cdot - 3 + a^{2} + a - 2 + 11 a = 2$

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

Rewrite using the commutative property of multiplication.

$2 \cdot 2 a \cdot a + 2 a \cdot - 3 - 3 (2 a) - 3 \cdot - 3 + a^{2} + a - 2 + 11 a = 2$

Step 2.2.3.1.2

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

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Step 2.2.3.1.2.1

Move $a$ .

$2 \cdot 2 (a \cdot a) + 2 a \cdot - 3 - 3 (2 a) - 3 \cdot - 3 + a^{2} + a - 2 + 11 a = 2$

Step 2.2.3.1.2.2

Multiply $a$ by $a$ .

$2 \cdot 2 a^{2} + 2 a \cdot - 3 - 3 (2 a) - 3 \cdot - 3 + a^{2} + a - 2 + 11 a = 2$

Step 2.2.3.1.3

Multiply $2$ by $2$ .

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

Step 2.2.3.1.4

Multiply $- 3$ by $2$ .

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

Step 2.2.3.1.5

Multiply $2$ by $- 3$ .

$4 a^{2} - 6 a - 6 a - 3 \cdot - 3 + a^{2} + a - 2 + 11 a = 2$

Step 2.2.3.1.6

Multiply $- 3$ by $- 3$ .

$4 a^{2} - 6 a - 6 a + 9 + a^{2} + a - 2 + 11 a = 2$

Step 2.2.3.2

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

$4 a^{2} - 12 a + 9 + a^{2} + a - 2 + 11 a = 2$

Step 2.3

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

$5 a^{2} - 12 a + 9 + a - 2 + 11 a = 2$

Step 2.4

Add $- 12 a$ and $a$ .

$5 a^{2} - 11 a + 9 - 2 + 11 a = 2$

Step 2.5

Combine the opposite terms in $5 a^{2} - 11 a + 9 - 2 + 11 a$ .

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Step 2.5.1

Add $- 11 a$ and $11 a$ .

$5 a^{2} + 0 + 9 - 2 = 2$

Step 2.5.2

Add $5 a^{2}$ and $0$ .

$5 a^{2} + 9 - 2 = 2$

Step 2.6

Subtract $2$ from $9$ .

$5 a^{2} + 7 = 2$

Step 3

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

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Step 3.1

Subtract $7$ from both sides of the equation.

$5 a^{2} = 2 - 7$

Step 3.2

Subtract $7$ from $2$ .

$5 a^{2} = - 5$

Step 4

Divide each term in $5 a^{2} = - 5$ by $5$ and simplify.

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Step 4.1

Divide each term in $5 a^{2} = - 5$ by $5$ .

$\frac{5 a^{2}}{5} = \frac{- 5}{5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $5$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{5 a^{2}}{5} = \frac{- 5}{5}$

Step 4.2.1.2

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

$a^{2} = \frac{- 5}{5}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 5$ by $5$ .

$a^{2} = - 1$

Step 5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$a = \pm \sqrt{- 1}$

Step 6

Rewrite $\sqrt{- 1}$ as $i$ .

$a = \pm i$

Step 7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.1

First, use the positive value of the $\pm$ to find the first solution.

$a = i$

Step 7.2

Next, use the negative value of the $\pm$ to find the second solution.

$a = - i$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

$a = i, - i$