Pre-Algebra Examples

$4 x + 2 + {(x + 1)}^{2} = {(3 x - 2)}^{2} + 3$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$4 x + 2 + {(x + 1)}^{2} = (3 x - 2) (3 x - 2) + 3$

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

$4 x + 2 + {(x + 1)}^{2} = 3 x (3 x - 2) - 2 (3 x - 2) + 3$

Step 1.1.2.2

Apply the distributive property.

$4 x + 2 + {(x + 1)}^{2} = 3 x (3 x) + 3 x \cdot - 2 - 2 (3 x - 2) + 3$

Step 1.1.2.3

Apply the distributive property.

$4 x + 2 + {(x + 1)}^{2} = 3 x (3 x) + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2 + 3$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Rewrite using the commutative property of multiplication.

$4 x + 2 + {(x + 1)}^{2} = 3 \cdot 3 x \cdot x + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2 + 3$

Step 1.1.3.1.2

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

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Step 1.1.3.1.2.1

Move $x$ .

$4 x + 2 + {(x + 1)}^{2} = 3 \cdot 3 (x \cdot x) + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2 + 3$

Step 1.1.3.1.2.2

Multiply $x$ by $x$ .

$4 x + 2 + {(x + 1)}^{2} = 3 \cdot 3 x^{2} + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2 + 3$

Step 1.1.3.1.3

Multiply $3$ by $3$ .

$4 x + 2 + {(x + 1)}^{2} = 9 x^{2} + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2 + 3$

Step 1.1.3.1.4

Multiply $- 2$ by $3$ .

$4 x + 2 + {(x + 1)}^{2} = 9 x^{2} - 6 x - 2 (3 x) - 2 \cdot - 2 + 3$

Step 1.1.3.1.5

Multiply $3$ by $- 2$ .

$4 x + 2 + {(x + 1)}^{2} = 9 x^{2} - 6 x - 6 x - 2 \cdot - 2 + 3$

Step 1.1.3.1.6

Multiply $- 2$ by $- 2$ .

$4 x + 2 + {(x + 1)}^{2} = 9 x^{2} - 6 x - 6 x + 4 + 3$

Step 1.1.3.2

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

$4 x + 2 + {(x + 1)}^{2} = 9 x^{2} - 12 x + 4 + 3$

Step 1.2

Add $4$ and $3$ .

$4 x + 2 + {(x + 1)}^{2} = 9 x^{2} - 12 x + 7$

Step 2

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

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Step 2.1

Subtract $9 x^{2}$ from both sides of the equation.

$4 x + 2 + {(x + 1)}^{2} - 9 x^{2} = - 12 x + 7$

Step 2.2

Add $12 x$ to both sides of the equation.

$4 x + 2 + {(x + 1)}^{2} - 9 x^{2} + 12 x = 7$

Step 2.3

Simplify each term.

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Step 2.3.1

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$4 x + 2 + (x + 1) (x + 1) - 9 x^{2} + 12 x = 7$

Step 2.3.2

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

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Step 2.3.2.1

Apply the distributive property.

$4 x + 2 + x (x + 1) + 1 (x + 1) - 9 x^{2} + 12 x = 7$

Step 2.3.2.2

Apply the distributive property.

$4 x + 2 + x \cdot x + x \cdot 1 + 1 (x + 1) - 9 x^{2} + 12 x = 7$

Step 2.3.2.3

Apply the distributive property.

$4 x + 2 + x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1 - 9 x^{2} + 12 x = 7$

Step 2.3.3

Simplify and combine like terms.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Multiply $x$ by $x$ .

$4 x + 2 + x^{2} + x \cdot 1 + 1 x + 1 \cdot 1 - 9 x^{2} + 12 x = 7$

Step 2.3.3.1.2

Multiply $x$ by $1$ .

$4 x + 2 + x^{2} + x + 1 x + 1 \cdot 1 - 9 x^{2} + 12 x = 7$

Step 2.3.3.1.3

Multiply $x$ by $1$ .

$4 x + 2 + x^{2} + x + x + 1 \cdot 1 - 9 x^{2} + 12 x = 7$

Step 2.3.3.1.4

Multiply $1$ by $1$ .

$4 x + 2 + x^{2} + x + x + 1 - 9 x^{2} + 12 x = 7$

Step 2.3.3.2

Add $x$ and $x$ .

$4 x + 2 + x^{2} + 2 x + 1 - 9 x^{2} + 12 x = 7$

Step 2.4

Add $4 x$ and $2 x$ .

$6 x + 2 + x^{2} + 1 - 9 x^{2} + 12 x = 7$

Step 2.5

Add $6 x$ and $12 x$ .

$18 x + 2 + x^{2} + 1 - 9 x^{2} = 7$

Step 2.6

Add $2$ and $1$ .

$18 x + x^{2} + 3 - 9 x^{2} = 7$

Step 2.7

Subtract $9 x^{2}$ from $x^{2}$ .

$18 x - 8 x^{2} + 3 = 7$

Step 3

Subtract $7$ from both sides of the equation.

$18 x - 8 x^{2} + 3 - 7 = 0$

Step 4

Subtract $7$ from $3$ .

$18 x - 8 x^{2} - 4 = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Factor $- 2$ out of $18 x - 8 x^{2} - 4$ .

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Step 5.1.1

Reorder $18 x$ and $- 8 x^{2}$ .

$- 8 x^{2} + 18 x - 4 = 0$

Step 5.1.2

Factor $- 2$ out of $- 8 x^{2}$ .

$- 2 (4 x^{2}) + 18 x - 4 = 0$

Step 5.1.3

Factor $- 2$ out of $18 x$ .

$- 2 (4 x^{2}) - 2 (- 9 x) - 4 = 0$

Step 5.1.4

Factor $- 2$ out of $- 4$ .

$- 2 (4 x^{2}) - 2 (- 9 x) - 2 \cdot 2 = 0$

Step 5.1.5

Factor $- 2$ out of $- 2 (4 x^{2}) - 2 (- 9 x)$ .

$- 2 (4 x^{2} - 9 x) - 2 \cdot 2 = 0$

Step 5.1.6

Factor $- 2$ out of $- 2 (4 x^{2} - 9 x) - 2 (2)$ .

$- 2 (4 x^{2} - 9 x + 2) = 0$

Step 5.2

Factor.

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Step 5.2.1

Factor by grouping.

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Step 5.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 2 = 8$ and whose sum is $b = - 9$ .

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Step 5.2.1.1.1

Factor $- 9$ out of $- 9 x$ .

$- 2 (4 x^{2} - 9 x + 2) = 0$

Step 5.2.1.1.2

Rewrite $- 9$ as $- 1$ plus $- 8$

$- 2 (4 x^{2} + (- 1 - 8) x + 2) = 0$

Step 5.2.1.1.3

Apply the distributive property.

$- 2 (4 x^{2} - 1 x - 8 x + 2) = 0$

Step 5.2.1.2

Factor out the greatest common factor from each group.

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Step 5.2.1.2.1

Group the first two terms and the last two terms.

$- 2 ((4 x^{2} - 1 x) - 8 x + 2) = 0$

Step 5.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- 2 (x (4 x - 1) - 2 (4 x - 1)) = 0$

Step 5.2.1.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 1$ .

$- 2 ((4 x - 1) (x - 2)) = 0$

Step 5.2.2

Remove unnecessary parentheses.

$- 2 (4 x - 1) (x - 2) = 0$

Step 6

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

$4 x - 1 = 0$

$x - 2 = 0$

Step 7

Set $4 x - 1$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Step 7.2

Solve $4 x - 1 = 0$ for $x$ .

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Step 7.2.1

Add $1$ to both sides of the equation.

$4 x = 1$

Step 7.2.2

Divide each term in $4 x = 1$ by $4$ and simplify.

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Step 7.2.2.1

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $4$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 8

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

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Step 8.1

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

$x - 2 = 0$

Step 8.2

Add $2$ to both sides of the equation.

$x = 2$

Step 9

The final solution is all the values that make $- 2 (4 x - 1) (x - 2) = 0$ true.

$x = \frac{1}{4}, 2$