Pre-Algebra Examples

$3 {(x - 6)}^{2} + x^{2} = x (x + 71) - 47 x$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x (x + 71) - 47 x = 3 {(x - 6)}^{2} + x^{2}$

Step 2

Simplify $x (x + 71) - 47 x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$x \cdot x + x \cdot 71 - 47 x = 3 {(x - 6)}^{2} + x^{2}$

Step 2.1.2

Multiply $x$ by $x$ .

$x^{2} + x \cdot 71 - 47 x = 3 {(x - 6)}^{2} + x^{2}$

Step 2.1.3

Move $71$ to the left of $x$ .

$x^{2} + 71 x - 47 x = 3 {(x - 6)}^{2} + x^{2}$

Step 2.2

Subtract $47 x$ from $71 x$ .

$x^{2} + 24 x = 3 {(x - 6)}^{2} + x^{2}$

Step 3

Simplify $3 {(x - 6)}^{2} + x^{2}$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$x^{2} + 24 x = 3 ((x - 6) (x - 6)) + x^{2}$

Step 3.1.2

Expand $(x - 6) (x - 6)$ using the FOIL Method.

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Step 3.1.2.1

Apply the distributive property.

$x^{2} + 24 x = 3 (x (x - 6) - 6 (x - 6)) + x^{2}$

Step 3.1.2.2

Apply the distributive property.

$x^{2} + 24 x = 3 (x \cdot x + x \cdot - 6 - 6 (x - 6)) + x^{2}$

Step 3.1.2.3

Apply the distributive property.

$x^{2} + 24 x = 3 (x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6) + x^{2}$

Step 3.1.3

Simplify and combine like terms.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + 24 x = 3 (x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6) + x^{2}$

Step 3.1.3.1.2

Move $- 6$ to the left of $x$ .

$x^{2} + 24 x = 3 (x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6) + x^{2}$

Step 3.1.3.1.3

Multiply $- 6$ by $- 6$ .

$x^{2} + 24 x = 3 (x^{2} - 6 x - 6 x + 36) + x^{2}$

Step 3.1.3.2

Subtract $6 x$ from $- 6 x$ .

$x^{2} + 24 x = 3 (x^{2} - 12 x + 36) + x^{2}$

Step 3.1.4

Apply the distributive property.

$x^{2} + 24 x = 3 x^{2} + 3 (- 12 x) + 3 \cdot 36 + x^{2}$

Step 3.1.5

Simplify.

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Step 3.1.5.1

Multiply $- 12$ by $3$ .

$x^{2} + 24 x = 3 x^{2} - 36 x + 3 \cdot 36 + x^{2}$

Step 3.1.5.2

Multiply $3$ by $36$ .

$x^{2} + 24 x = 3 x^{2} - 36 x + 108 + x^{2}$

Step 3.2

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

$x^{2} + 24 x = 4 x^{2} - 36 x + 108$

Step 4

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

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Step 4.1

Subtract $4 x^{2}$ from both sides of the equation.

$x^{2} + 24 x - 4 x^{2} = - 36 x + 108$

Step 4.2

Add $36 x$ to both sides of the equation.

$x^{2} + 24 x - 4 x^{2} + 36 x = 108$

Step 4.3

Subtract $4 x^{2}$ from $x^{2}$ .

$- 3 x^{2} + 24 x + 36 x = 108$

Step 4.4

Add $24 x$ and $36 x$ .

$- 3 x^{2} + 60 x = 108$

Step 5

Subtract $108$ from both sides of the equation.

$- 3 x^{2} + 60 x - 108 = 0$

Step 6

Factor the left side of the equation.

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Step 6.1

Factor $- 3$ out of $- 3 x^{2} + 60 x - 108$ .

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Step 6.1.1

Factor $- 3$ out of $- 3 x^{2}$ .

$- 3 x^{2} + 60 x - 108 = 0$

Step 6.1.2

Factor $- 3$ out of $60 x$ .

$- 3 x^{2} - 3 (- 20 x) - 108 = 0$

Step 6.1.3

Factor $- 3$ out of $- 108$ .

$- 3 x^{2} - 3 (- 20 x) - 3 \cdot 36 = 0$

Step 6.1.4

Factor $- 3$ out of $- 3 (x^{2}) - 3 (- 20 x)$ .

$- 3 (x^{2} - 20 x) - 3 \cdot 36 = 0$

Step 6.1.5

Factor $- 3$ out of $- 3 (x^{2} - 20 x) - 3 (36)$ .

$- 3 (x^{2} - 20 x + 36) = 0$

Step 6.2

Factor.

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Step 6.2.1

Factor $x^{2} - 20 x + 36$ using the AC method.

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Step 6.2.1.1

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 $36$ and whose sum is $- 20$ .

$- 18, - 2$

Step 6.2.1.2

Write the factored form using these integers.

$- 3 ((x - 18) (x - 2)) = 0$

Step 6.2.2

Remove unnecessary parentheses.

$- 3 (x - 18) (x - 2) = 0$

Step 7

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

$x - 18 = 0$

$x - 2 = 0$

Step 8

Set $x - 18$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x - 18$ equal to $0$ .

$x - 18 = 0$

Step 8.2

Add $18$ to both sides of the equation.

$x = 18$

Step 9

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 9.2

Add $2$ to both sides of the equation.

$x = 2$

Step 10

The final solution is all the values that make $- 3 (x - 18) (x - 2) = 0$ true.

$x = 18, 2$