Finite Math Examples

$x (x + 2) + 5 = 3 (2 - x) + x - 4$

Step 1

Simplify the equation into a proper form to complete the square.

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$x \cdot x + x \cdot 2 + 5 = 3 (2 - x) + x - 4$

Step 1.1.2

Multiply $x$ by $x$ .

$x^{2} + x \cdot 2 + 5 = 3 (2 - x) + x - 4$

Step 1.1.3

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

$x^{2} + 2 x + 5 = 3 (2 - x) + x - 4$

Step 1.2

Simplify $3 (2 - x) + x - 4$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Apply the distributive property.

$x^{2} + 2 x + 5 = 3 \cdot 2 + 3 (- x) + x - 4$

Step 1.2.1.2

Multiply $3$ by $2$ .

$x^{2} + 2 x + 5 = 6 + 3 (- x) + x - 4$

Step 1.2.1.3

Multiply $- 1$ by $3$ .

$x^{2} + 2 x + 5 = 6 - 3 x + x - 4$

Step 1.2.2

Simplify by adding terms.

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Step 1.2.2.1

Subtract $4$ from $6$ .

$x^{2} + 2 x + 5 = - 3 x + x + 2$

Step 1.2.2.2

Add $- 3 x$ and $x$ .

$x^{2} + 2 x + 5 = - 2 x + 2$

Step 1.3

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

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Step 1.3.1

Subtract $5$ from both sides of the equation.

$x^{2} + 2 x = - 2 x + 2 - 5$

Step 1.3.2

Subtract $5$ from $2$ .

$x^{2} + 2 x = - 2 x - 3$

Step 1.4

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

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Step 1.4.1

Add $2 x$ to both sides of the equation.

$x^{2} + 2 x + 2 x = - 3$

Step 1.4.2

Add $2 x$ and $2 x$ .

$x^{2} + 4 x = - 3$

Step 2

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(2)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + 4 x + {(2)}^{2} = - 3 + {(2)}^{2}$

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

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

$x^{2} + 4 x + 4 = - 3 + {(2)}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

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

$x^{2} + 4 x + 4 = - 3 + 4$

Step 4.2.1.2

Add $- 3$ and $4$ .

$x^{2} + 4 x + 4 = 1$

Step 5

Factor the perfect trinomial square into ${(x + 2)}^{2}$ .

${(x + 2)}^{2} = 1$

Step 6

Solve the equation for $x$ .

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Step 6.1

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

$x + 2 = \pm \sqrt{1}$

Step 6.2

Any root of $1$ is $1$ .

$x + 2 = \pm 1$

Step 6.3

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

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Step 6.3.1

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

$x + 2 = 1$

Step 6.3.2

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

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Step 6.3.2.1

Subtract $2$ from both sides of the equation.

$x = 1 - 2$

Step 6.3.2.2

Subtract $2$ from $1$ .

$x = - 1$

Step 6.3.3

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

$x + 2 = - 1$

Step 6.3.4

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

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Step 6.3.4.1

Subtract $2$ from both sides of the equation.

$x = - 1 - 2$

Step 6.3.4.2

Subtract $2$ from $- 1$ .

$x = - 3$

Step 6.3.5

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

$x = - 1, - 3$