Algebra Examples

${(3 x - 1)}^{2} - 5 (x - 2) - {(2 x + 3)}^{2} - (5 x + 2) (x - 1) = 0$

Step 1

Simplify ${(3 x - 1)}^{2} - 5 (x - 2) - {(2 x + 3)}^{2} - (5 x + 2) (x - 1)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

${(3 x - 1)}^{2} - 5 x - 5 \cdot - 2 - {(2 x + 3)}^{2} - (5 x + 2) (x - 1) = 0$

Step 1.1.2

Multiply $- 5$ by $- 2$ .

${(3 x - 1)}^{2} - 5 x + 10 - {(2 x + 3)}^{2} - (5 x + 2) (x - 1) = 0$

Step 1.1.3

Apply the distributive property.

${(3 x - 1)}^{2} - 5 x + 10 - {(2 x + 3)}^{2} + (- (5 x) - 1 \cdot 2) (x - 1) = 0$

Step 1.1.4

Multiply $5$ by $- 1$ .

${(3 x - 1)}^{2} - 5 x + 10 - {(2 x + 3)}^{2} + (- 5 x - 1 \cdot 2) (x - 1) = 0$

Step 1.1.5

Multiply $- 1$ by $2$ .

${(3 x - 1)}^{2} - 5 x + 10 - {(2 x + 3)}^{2} + (- 5 x - 2) (x - 1) = 0$

Step 1.1.6

Expand $(- 5 x - 2) (x - 1)$ using the FOIL Method.

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Step 1.1.6.1

Apply the distributive property.

${(3 x - 1)}^{2} - 5 x + 10 - {(2 x + 3)}^{2} - 5 x (x - 1) - 2 (x - 1) = 0$

Step 1.1.6.2

Apply the distributive property.

${(3 x - 1)}^{2} - 5 x + 10 - {(2 x + 3)}^{2} - 5 x \cdot x - 5 x \cdot - 1 - 2 (x - 1) = 0$

Step 1.1.6.3

Apply the distributive property.

${(3 x - 1)}^{2} - 5 x + 10 - {(2 x + 3)}^{2} - 5 x \cdot x - 5 x \cdot - 1 - 2 x - 2 \cdot - 1 = 0$

Step 1.1.7

Simplify and combine like terms.

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Step 1.1.7.1

Simplify each term.

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Step 1.1.7.1.1

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

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Step 1.1.7.1.1.1

Move $x$ .

${(3 x - 1)}^{2} - 5 x + 10 - {(2 x + 3)}^{2} - 5 (x \cdot x) - 5 x \cdot - 1 - 2 x - 2 \cdot - 1 = 0$

Step 1.1.7.1.1.2

Multiply $x$ by $x$ .

${(3 x - 1)}^{2} - 5 x + 10 - {(2 x + 3)}^{2} - 5 x^{2} - 5 x \cdot - 1 - 2 x - 2 \cdot - 1 = 0$

Step 1.1.7.1.2

Multiply $- 1$ by $- 5$ .

${(3 x - 1)}^{2} - 5 x + 10 - {(2 x + 3)}^{2} - 5 x^{2} + 5 x - 2 x - 2 \cdot - 1 = 0$

Step 1.1.7.1.3

Multiply $- 2$ by $- 1$ .

${(3 x - 1)}^{2} - 5 x + 10 - {(2 x + 3)}^{2} - 5 x^{2} + 5 x - 2 x + 2 = 0$

Step 1.1.7.2

Subtract $2 x$ from $5 x$ .

${(3 x - 1)}^{2} - 5 x + 10 - {(2 x + 3)}^{2} - 5 x^{2} + 3 x + 2 = 0$

Step 1.2

Simplify by adding terms.

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Step 1.2.1

Add $- 5 x$ and $3 x$ .

${(3 x - 1)}^{2} + 10 - {(2 x + 3)}^{2} - 5 x^{2} - 2 x + 2 = 0$

Step 1.2.2

Add $10$ and $2$ .

${(3 x - 1)}^{2} - {(2 x + 3)}^{2} - 5 x^{2} - 2 x + 12 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Regroup terms.

$- 2 x + 12 + {(3 x - 1)}^{2} - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.2

Factor $- 2$ out of $- 2 x + 12$ .

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Step 2.2.1

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

$- 2 x + 12 + {(3 x - 1)}^{2} - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.2.2

Factor $- 2$ out of $12$ .

$- 2 x - 2 \cdot - 6 + {(3 x - 1)}^{2} - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.2.3

Factor $- 2$ out of $- 2 (x) - 2 (- 6)$ .

$- 2 (x - 6) + {(3 x - 1)}^{2} - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.3

Rewrite ${(3 x - 1)}^{2}$ as $(3 x - 1) (3 x - 1)$ .

$- 2 (x - 6) + (3 x - 1) (3 x - 1) - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.4

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

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Step 2.4.1

Apply the distributive property.

$- 2 (x - 6) + 3 x (3 x - 1) - 1 (3 x - 1) - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.4.2

Apply the distributive property.

$- 2 (x - 6) + 3 x (3 x) + 3 x \cdot - 1 - 1 (3 x - 1) - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.4.3

Apply the distributive property.

$- 2 (x - 6) + 3 x (3 x) + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.5

Simplify and combine like terms.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Rewrite using the commutative property of multiplication.

$- 2 (x - 6) + 3 \cdot (3 x \cdot x) + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.5.1.2

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

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Step 2.5.1.2.1

Move $x$ .

$- 2 (x - 6) + 3 \cdot (3 (x \cdot x)) + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.5.1.2.2

Multiply $x$ by $x$ .

$- 2 (x - 6) + 3 \cdot (3 x^{2}) + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.5.1.3

Multiply $3$ by $3$ .

$- 2 (x - 6) + 9 x^{2} + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1 - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.5.1.4

Multiply $- 1$ by $3$ .

$- 2 (x - 6) + 9 x^{2} - 3 x - 1 (3 x) - 1 \cdot - 1 - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.5.1.5

Multiply $3$ by $- 1$ .

$- 2 (x - 6) + 9 x^{2} - 3 x - 3 x - 1 \cdot - 1 - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.5.1.6

Multiply $- 1$ by $- 1$ .

$- 2 (x - 6) + 9 x^{2} - 3 x - 3 x + 1 - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.5.2

Subtract $3 x$ from $- 3 x$ .

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - {(2 x + 3)}^{2} - 5 x^{2} = 0$

Step 2.6

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

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - ((2 x + 3) (2 x + 3)) - 5 x^{2} = 0$

Step 2.7

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

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Step 2.7.1

Apply the distributive property.

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - (2 x (2 x + 3) + 3 (2 x + 3)) - 5 x^{2} = 0$

Step 2.7.2

Apply the distributive property.

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - (2 x (2 x) + 2 x \cdot 3 + 3 (2 x + 3)) - 5 x^{2} = 0$

Step 2.7.3

Apply the distributive property.

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - (2 x (2 x) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3) - 5 x^{2} = 0$

Step 2.8

Simplify and combine like terms.

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

Rewrite using the commutative property of multiplication.

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - (2 \cdot (2 x \cdot x) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3) - 5 x^{2} = 0$

Step 2.8.1.2

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

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Step 2.8.1.2.1

Move $x$ .

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - (2 \cdot (2 (x \cdot x)) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3) - 5 x^{2} = 0$

Step 2.8.1.2.2

Multiply $x$ by $x$ .

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - (2 \cdot (2 x^{2}) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3) - 5 x^{2} = 0$

Step 2.8.1.3

Multiply $2$ by $2$ .

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - (4 x^{2} + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3) - 5 x^{2} = 0$

Step 2.8.1.4

Multiply $3$ by $2$ .

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - (4 x^{2} + 6 x + 3 (2 x) + 3 \cdot 3) - 5 x^{2} = 0$

Step 2.8.1.5

Multiply $2$ by $3$ .

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - (4 x^{2} + 6 x + 6 x + 3 \cdot 3) - 5 x^{2} = 0$

Step 2.8.1.6

Multiply $3$ by $3$ .

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - (4 x^{2} + 6 x + 6 x + 9) - 5 x^{2} = 0$

Step 2.8.2

Add $6 x$ and $6 x$ .

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - (4 x^{2} + 12 x + 9) - 5 x^{2} = 0$

Step 2.9

Apply the distributive property.

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - (4 x^{2}) - (12 x) - 1 \cdot 9 - 5 x^{2} = 0$

Step 2.10

Simplify.

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Step 2.10.1

Multiply $4$ by $- 1$ .

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - 4 x^{2} - (12 x) - 1 \cdot 9 - 5 x^{2} = 0$

Step 2.10.2

Multiply $12$ by $- 1$ .

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - 4 x^{2} - 12 x - 1 \cdot 9 - 5 x^{2} = 0$

Step 2.10.3

Multiply $- 1$ by $9$ .

$- 2 (x - 6) + 9 x^{2} - 6 x + 1 - 4 x^{2} - 12 x - 9 - 5 x^{2} = 0$

Step 2.11

Subtract $4 x^{2}$ from $9 x^{2}$ .

$- 2 (x - 6) + 5 x^{2} - 6 x + 1 - 12 x - 9 - 5 x^{2} = 0$

Step 2.12

Subtract $5 x^{2}$ from $5 x^{2}$ .

$- 2 (x - 6) - 6 x + 1 - 12 x - 9 + 0 = 0$

Step 2.13

Add $- 6 x + 1 - 12 x - 9$ and $0$ .

$- 2 (x - 6) - 6 x + 1 - 12 x - 9 = 0$

Step 2.14

Subtract $12 x$ from $- 6 x$ .

$- 2 (x - 6) - 18 x + 1 - 9 = 0$

Step 2.15

Subtract $9$ from $1$ .

$- 2 (x - 6) - 18 x - 8 = 0$

Step 2.16

Factor $2$ out of $- 18 x - 8$ .

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Step 2.16.1

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

$- 2 (x - 6) + 2 (- 9 x) - 8 = 0$

Step 2.16.2

Factor $2$ out of $- 8$ .

$- 2 (x - 6) + 2 (- 9 x) + 2 (- 4) = 0$

Step 2.16.3

Factor $2$ out of $2 (- 9 x) + 2 (- 4)$ .

$- 2 (x - 6) + 2 (- 9 x - 4) = 0$

Step 2.17

Factor $2$ out of $- 2 (x - 6) + 2 (- 9 x - 4)$ .

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Step 2.17.1

Factor $2$ out of $- 2 (x - 6)$ .

$2 (- (x - 6)) + 2 (- 9 x - 4) = 0$

Step 2.17.2

Factor $2$ out of $2 (- (x - 6)) + 2 (- 9 x - 4)$ .

$2 (- (x - 6) - 9 x - 4) = 0$

Step 2.18

Apply the distributive property.

$2 (- x + 6 - 9 x - 4) = 0$

Step 2.19

Multiply $- 1$ by $- 6$ .

$2 (- x + 6 - 9 x - 4) = 0$

Step 2.20

Subtract $9 x$ from $- x$ .

$2 (- 10 x + 6 - 4) = 0$

Step 2.21

Subtract $4$ from $6$ .

$2 (- 10 x + 2) = 0$

Step 2.22

Factor.

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Step 2.22.1

Factor $2$ out of $- 10 x + 2$ .

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Step 2.22.1.1

Factor $2$ out of $- 10 x$ .

$2 (2 (- 5 x) + 2) = 0$

Step 2.22.1.2

Factor $2$ out of $2$ .

$2 (2 (- 5 x) + 2 (1)) = 0$

Step 2.22.1.3

Factor $2$ out of $2 (- 5 x) + 2 (1)$ .

$2 (2 (- 5 x + 1)) = 0$

Step 2.22.2

Remove unnecessary parentheses.

$2 \cdot (2 (- 5 x + 1)) = 0$

Step 2.23

Multiply $2$ by $2$ .

$4 (- 5 x + 1) = 0$

Step 3

Divide each term in $4 (- 5 x + 1) = 0$ by $4$ and simplify.

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Step 3.1

Divide each term in $4 (- 5 x + 1) = 0$ by $4$ .

$\frac{4 (- 5 x + 1)}{4} = \frac{0}{4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4 (- 5 x + 1)}{4} = \frac{0}{4}$

Step 3.2.1.2

Divide $- 5 x + 1$ by $1$ .

$- 5 x + 1 = \frac{0}{4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $4$ .

$- 5 x + 1 = 0$

Step 4

Subtract $1$ from both sides of the equation.

$- 5 x = - 1$

Step 5

Divide each term in $- 5 x = - 1$ by $- 5$ and simplify.

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Step 5.1

Divide each term in $- 5 x = - 1$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 1}{- 5}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 5$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 1}{- 5}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 5}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{5}$

Decimal Form:

$x = 0.2$