Pre-Algebra Examples

$- 3 x (4 x - 9) + 9 x - 43 = 2 (x + 9)$

Step 1

Simplify $- 3 x (4 x - 9) + 9 x - 43$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$- 3 x (4 x) - 3 x \cdot - 9 + 9 x - 43 = 2 (x + 9)$

Step 1.1.2

Rewrite using the commutative property of multiplication.

$- 3 \cdot 4 x \cdot x - 3 x \cdot - 9 + 9 x - 43 = 2 (x + 9)$

Step 1.1.3

Multiply $- 9$ by $- 3$ .

$- 3 \cdot 4 x \cdot x + 27 x + 9 x - 43 = 2 (x + 9)$

Step 1.1.4

Simplify each term.

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Step 1.1.4.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.4.1.1

Move $x$ .

$- 3 \cdot 4 (x \cdot x) + 27 x + 9 x - 43 = 2 (x + 9)$

Step 1.1.4.1.2

Multiply $x$ by $x$ .

$- 3 \cdot 4 x^{2} + 27 x + 9 x - 43 = 2 (x + 9)$

Step 1.1.4.2

Multiply $- 3$ by $4$ .

$- 12 x^{2} + 27 x + 9 x - 43 = 2 (x + 9)$

Step 1.2

Add $27 x$ and $9 x$ .

$- 12 x^{2} + 36 x - 43 = 2 (x + 9)$

Step 2

Simplify $2 (x + 9)$ .

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Step 2.1

Apply the distributive property.

$- 12 x^{2} + 36 x - 43 = 2 x + 2 \cdot 9$

Step 2.2

Multiply $2$ by $9$ .

$- 12 x^{2} + 36 x - 43 = 2 x + 18$

Step 3

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

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Step 3.1

Subtract $2 x$ from both sides of the equation.

$- 12 x^{2} + 36 x - 43 - 2 x = 18$

Step 3.2

Subtract $2 x$ from $36 x$ .

$- 12 x^{2} + 34 x - 43 = 18$

Step 4

Move all terms to the left side of the equation and simplify.

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Step 4.1

Subtract $18$ from both sides of the equation.

$- 12 x^{2} + 34 x - 43 - 18 = 0$

Step 4.2

Subtract $18$ from $- 43$ .

$- 12 x^{2} + 34 x - 61 = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = - 12$ , $b = 34$ , and $c = - 61$ into the quadratic formula and solve for $x$ .

$\frac{- 34 \pm \sqrt{34^{2} - 4 \cdot (- 12 \cdot - 61)}}{2 \cdot - 12}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $34$ to the power of $2$ .

$x = \frac{- 34 \pm \sqrt{1156 - 4 \cdot - 12 \cdot - 61}}{2 \cdot - 12}$

Step 7.1.2

Multiply $- 4 \cdot - 12 \cdot - 61$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 12$ .

$x = \frac{- 34 \pm \sqrt{1156 + 48 \cdot - 61}}{2 \cdot - 12}$

Step 7.1.2.2

Multiply $48$ by $- 61$ .

$x = \frac{- 34 \pm \sqrt{1156 - 2928}}{2 \cdot - 12}$

Step 7.1.3

Subtract $2928$ from $1156$ .

$x = \frac{- 34 \pm \sqrt{- 1772}}{2 \cdot - 12}$

Step 7.1.4

Rewrite $- 1772$ as $- 1 (1772)$ .

$x = \frac{- 34 \pm \sqrt{- 1 \cdot 1772}}{2 \cdot - 12}$

Step 7.1.5

Rewrite $\sqrt{- 1 (1772)}$ as $\sqrt{- 1} \cdot \sqrt{1772}$ .

$x = \frac{- 34 \pm \sqrt{- 1} \cdot \sqrt{1772}}{2 \cdot - 12}$

Step 7.1.6

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

$x = \frac{- 34 \pm i \cdot \sqrt{1772}}{2 \cdot - 12}$

Step 7.1.7

Rewrite $1772$ as $2^{2} \cdot 443$ .

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Step 7.1.7.1

Factor $4$ out of $1772$ .

$x = \frac{- 34 \pm i \cdot \sqrt{4 (443)}}{2 \cdot - 12}$

Step 7.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 34 \pm i \cdot \sqrt{2^{2} \cdot 443}}{2 \cdot - 12}$

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{- 34 \pm i \cdot (2 \sqrt{443})}{2 \cdot - 12}$

Step 7.1.9

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

$x = \frac{- 34 \pm 2 i \sqrt{443}}{2 \cdot - 12}$

Step 7.2

Multiply $2$ by $- 12$ .

$x = \frac{- 34 \pm 2 i \sqrt{443}}{- 24}$

Step 7.3

Simplify $\frac{- 34 \pm 2 i \sqrt{443}}{- 24}$ .

$x = \frac{17 \pm i \sqrt{443}}{12}$

Step 8

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $34$ to the power of $2$ .

$x = \frac{- 34 \pm \sqrt{1156 - 4 \cdot - 12 \cdot - 61}}{2 \cdot - 12}$

Step 8.1.2

Multiply $- 4 \cdot - 12 \cdot - 61$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 12$ .

$x = \frac{- 34 \pm \sqrt{1156 + 48 \cdot - 61}}{2 \cdot - 12}$

Step 8.1.2.2

Multiply $48$ by $- 61$ .

$x = \frac{- 34 \pm \sqrt{1156 - 2928}}{2 \cdot - 12}$

Step 8.1.3

Subtract $2928$ from $1156$ .

$x = \frac{- 34 \pm \sqrt{- 1772}}{2 \cdot - 12}$

Step 8.1.4

Rewrite $- 1772$ as $- 1 (1772)$ .

$x = \frac{- 34 \pm \sqrt{- 1 \cdot 1772}}{2 \cdot - 12}$

Step 8.1.5

Rewrite $\sqrt{- 1 (1772)}$ as $\sqrt{- 1} \cdot \sqrt{1772}$ .

$x = \frac{- 34 \pm \sqrt{- 1} \cdot \sqrt{1772}}{2 \cdot - 12}$

Step 8.1.6

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

$x = \frac{- 34 \pm i \cdot \sqrt{1772}}{2 \cdot - 12}$

Step 8.1.7

Rewrite $1772$ as $2^{2} \cdot 443$ .

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Step 8.1.7.1

Factor $4$ out of $1772$ .

$x = \frac{- 34 \pm i \cdot \sqrt{4 (443)}}{2 \cdot - 12}$

Step 8.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 34 \pm i \cdot \sqrt{2^{2} \cdot 443}}{2 \cdot - 12}$

Step 8.1.8

Pull terms out from under the radical.

$x = \frac{- 34 \pm i \cdot (2 \sqrt{443})}{2 \cdot - 12}$

Step 8.1.9

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

$x = \frac{- 34 \pm 2 i \sqrt{443}}{2 \cdot - 12}$

Step 8.2

Multiply $2$ by $- 12$ .

$x = \frac{- 34 \pm 2 i \sqrt{443}}{- 24}$

Step 8.3

Simplify $\frac{- 34 \pm 2 i \sqrt{443}}{- 24}$ .

$x = \frac{17 \pm i \sqrt{443}}{12}$

Step 8.4

Change the $\pm$ to $+$ .

$x = \frac{17 + i \sqrt{443}}{12}$

Step 9

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $34$ to the power of $2$ .

$x = \frac{- 34 \pm \sqrt{1156 - 4 \cdot - 12 \cdot - 61}}{2 \cdot - 12}$

Step 9.1.2

Multiply $- 4 \cdot - 12 \cdot - 61$ .

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Step 9.1.2.1

Multiply $- 4$ by $- 12$ .

$x = \frac{- 34 \pm \sqrt{1156 + 48 \cdot - 61}}{2 \cdot - 12}$

Step 9.1.2.2

Multiply $48$ by $- 61$ .

$x = \frac{- 34 \pm \sqrt{1156 - 2928}}{2 \cdot - 12}$

Step 9.1.3

Subtract $2928$ from $1156$ .

$x = \frac{- 34 \pm \sqrt{- 1772}}{2 \cdot - 12}$

Step 9.1.4

Rewrite $- 1772$ as $- 1 (1772)$ .

$x = \frac{- 34 \pm \sqrt{- 1 \cdot 1772}}{2 \cdot - 12}$

Step 9.1.5

Rewrite $\sqrt{- 1 (1772)}$ as $\sqrt{- 1} \cdot \sqrt{1772}$ .

$x = \frac{- 34 \pm \sqrt{- 1} \cdot \sqrt{1772}}{2 \cdot - 12}$

Step 9.1.6

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

$x = \frac{- 34 \pm i \cdot \sqrt{1772}}{2 \cdot - 12}$

Step 9.1.7

Rewrite $1772$ as $2^{2} \cdot 443$ .

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Step 9.1.7.1

Factor $4$ out of $1772$ .

$x = \frac{- 34 \pm i \cdot \sqrt{4 (443)}}{2 \cdot - 12}$

Step 9.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 34 \pm i \cdot \sqrt{2^{2} \cdot 443}}{2 \cdot - 12}$

Step 9.1.8

Pull terms out from under the radical.

$x = \frac{- 34 \pm i \cdot (2 \sqrt{443})}{2 \cdot - 12}$

Step 9.1.9

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

$x = \frac{- 34 \pm 2 i \sqrt{443}}{2 \cdot - 12}$

Step 9.2

Multiply $2$ by $- 12$ .

$x = \frac{- 34 \pm 2 i \sqrt{443}}{- 24}$

Step 9.3

Simplify $\frac{- 34 \pm 2 i \sqrt{443}}{- 24}$ .

$x = \frac{17 \pm i \sqrt{443}}{12}$

Step 9.4

Change the $\pm$ to $-$ .

$x = \frac{17 - i \sqrt{443}}{12}$

Step 10

The final answer is the combination of both solutions.

$x = \frac{17 + i \sqrt{443}}{12}, \frac{17 - i \sqrt{443}}{12}$