Precalculus Examples

$2 x^{2} - 7 x + 9 = (x - 3) (x + 1) + 3 x$

Step 1

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

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Step 1.1

Simplify $(x - 3) (x + 1) + 3 x$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Expand $(x - 3) (x + 1)$ using the FOIL Method.

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Step 1.1.1.1.1

Apply the distributive property.

$2 x^{2} - 7 x + 9 = x (x + 1) - 3 (x + 1) + 3 x$

Step 1.1.1.1.2

Apply the distributive property.

$2 x^{2} - 7 x + 9 = x \cdot x + x \cdot 1 - 3 (x + 1) + 3 x$

Step 1.1.1.1.3

Apply the distributive property.

$2 x^{2} - 7 x + 9 = x \cdot x + x \cdot 1 - 3 x - 3 \cdot 1 + 3 x$

Step 1.1.1.2

Simplify and combine like terms.

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Step 1.1.1.2.1

Simplify each term.

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Step 1.1.1.2.1.1

Multiply $x$ by $x$ .

$2 x^{2} - 7 x + 9 = x^{2} + x \cdot 1 - 3 x - 3 \cdot 1 + 3 x$

Step 1.1.1.2.1.2

Multiply $x$ by $1$ .

$2 x^{2} - 7 x + 9 = x^{2} + x - 3 x - 3 \cdot 1 + 3 x$

Step 1.1.1.2.1.3

Multiply $- 3$ by $1$ .

$2 x^{2} - 7 x + 9 = x^{2} + x - 3 x - 3 + 3 x$

Step 1.1.1.2.2

Subtract $3 x$ from $x$ .

$2 x^{2} - 7 x + 9 = x^{2} - 2 x - 3 + 3 x$

Step 1.1.2

Add $- 2 x$ and $3 x$ .

$2 x^{2} - 7 x + 9 = x^{2} + x - 3$

Step 1.2

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

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Step 1.2.1

Subtract $9$ from both sides of the equation.

$2 x^{2} - 7 x = x^{2} + x - 3 - 9$

Step 1.2.2

Subtract $9$ from $- 3$ .

$2 x^{2} - 7 x = x^{2} + x - 12$

Step 1.3

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

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Step 1.3.1

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

$2 x^{2} - 7 x - x^{2} = x - 12$

Step 1.3.2

Subtract $x$ from both sides of the equation.

$2 x^{2} - 7 x - x^{2} - x = - 12$

Step 1.3.3

Subtract $x^{2}$ from $2 x^{2}$ .

$x^{2} - 7 x - x = - 12$

Step 1.3.4

Subtract $x$ from $- 7 x$ .

$x^{2} - 8 x = - 12$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- 4)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - 8 x + {(- 4)}^{2} = - 12 + {(- 4)}^{2}$

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 $- 4$ to the power of $2$ .

$x^{2} - 8 x + 16 = - 12 + {(- 4)}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 12 + {(- 4)}^{2}$ .

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Step 4.2.1.1

Raise $- 4$ to the power of $2$ .

$x^{2} - 8 x + 16 = - 12 + 16$

Step 4.2.1.2

Add $- 12$ and $16$ .

$x^{2} - 8 x + 16 = 4$

Step 5

Factor the perfect trinomial square into ${(x - 4)}^{2}$ .

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

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.

$x - 4 = \pm \sqrt{4}$

Step 6.2

Simplify $\pm \sqrt{4}$ .

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Step 6.2.1

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

$x - 4 = \pm \sqrt{2^{2}}$

Step 6.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x - 4 = \pm 2$

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.

$x - 4 = 2$

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

Add $4$ to both sides of the equation.

$x = 2 + 4$

Step 6.3.2.2

Add $2$ and $4$ .

$x = 6$

Step 6.3.3

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

$x - 4 = - 2$

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

Add $4$ to both sides of the equation.

$x = - 2 + 4$

Step 6.3.4.2

Add $- 2$ and $4$ .

$x = 2$

Step 6.3.5

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

$x = 6, 2$