Algebra Examples

$5 {(1 - x)}^{2} - 6 (x^{2} - 3 x - 7) = x (x - 3) - 2 x (x + 5) - 2$

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.

$x (x - 3) - 2 x (x + 5) - 2 = 5 {(1 - x)}^{2} - 6 (x^{2} - 3 x - 7)$

Step 2

Simplify $x (x - 3) - 2 x (x + 5) - 2$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 2 x (x + 5) - 2 = 5 {(1 - x)}^{2} - 6 (x^{2} - 3 x - 7)$

Step 2.1.2

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 3 - 2 x (x + 5) - 2 = 5 {(1 - x)}^{2} - 6 (x^{2} - 3 x - 7)$

Step 2.1.3

Move $- 3$ to the left of $x$ .

$x^{2} - 3 \cdot x - 2 x (x + 5) - 2 = 5 {(1 - x)}^{2} - 6 (x^{2} - 3 x - 7)$

Step 2.1.4

Apply the distributive property.

$x^{2} - 3 x - 2 x \cdot x - 2 x \cdot 5 - 2 = 5 {(1 - x)}^{2} - 6 (x^{2} - 3 x - 7)$

Step 2.1.5

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

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Step 2.1.5.1

Move $x$ .

$x^{2} - 3 x - 2 (x \cdot x) - 2 x \cdot 5 - 2 = 5 {(1 - x)}^{2} - 6 (x^{2} - 3 x - 7)$

Step 2.1.5.2

Multiply $x$ by $x$ .

$x^{2} - 3 x - 2 x^{2} - 2 x \cdot 5 - 2 = 5 {(1 - x)}^{2} - 6 (x^{2} - 3 x - 7)$

Step 2.1.6

Multiply $5$ by $- 2$ .

$x^{2} - 3 x - 2 x^{2} - 10 x - 2 = 5 {(1 - x)}^{2} - 6 (x^{2} - 3 x - 7)$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

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

$- x^{2} - 3 x - 10 x - 2 = 5 {(1 - x)}^{2} - 6 (x^{2} - 3 x - 7)$

Step 2.2.2

Subtract $10 x$ from $- 3 x$ .

$- x^{2} - 13 x - 2 = 5 {(1 - x)}^{2} - 6 (x^{2} - 3 x - 7)$

Step 3

Simplify $5 {(1 - x)}^{2} - 6 (x^{2} - 3 x - 7)$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

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

$- x^{2} - 13 x - 2 = 5 ((1 - x) (1 - x)) - 6 (x^{2} - 3 x - 7)$

Step 3.1.2

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

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Step 3.1.2.1

Apply the distributive property.

$- x^{2} - 13 x - 2 = 5 (1 (1 - x) - x (1 - x)) - 6 (x^{2} - 3 x - 7)$

Step 3.1.2.2

Apply the distributive property.

$- x^{2} - 13 x - 2 = 5 (1 \cdot 1 + 1 (- x) - x (1 - x)) - 6 (x^{2} - 3 x - 7)$

Step 3.1.2.3

Apply the distributive property.

$- x^{2} - 13 x - 2 = 5 (1 \cdot 1 + 1 (- x) - x \cdot 1 - x (- x)) - 6 (x^{2} - 3 x - 7)$

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 $1$ by $1$ .

$- x^{2} - 13 x - 2 = 5 (1 + 1 (- x) - x \cdot 1 - x (- x)) - 6 (x^{2} - 3 x - 7)$

Step 3.1.3.1.2

Multiply $- x$ by $1$ .

$- x^{2} - 13 x - 2 = 5 (1 - x - x \cdot 1 - x (- x)) - 6 (x^{2} - 3 x - 7)$

Step 3.1.3.1.3

Multiply $- 1$ by $1$ .

$- x^{2} - 13 x - 2 = 5 (1 - x - x - x (- x)) - 6 (x^{2} - 3 x - 7)$

Step 3.1.3.1.4

Rewrite using the commutative property of multiplication.

$- x^{2} - 13 x - 2 = 5 (1 - x - x - 1 \cdot - 1 x \cdot x) - 6 (x^{2} - 3 x - 7)$

Step 3.1.3.1.5

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

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Step 3.1.3.1.5.1

Move $x$ .

$- x^{2} - 13 x - 2 = 5 (1 - x - x - 1 \cdot - 1 (x \cdot x)) - 6 (x^{2} - 3 x - 7)$

Step 3.1.3.1.5.2

Multiply $x$ by $x$ .

$- x^{2} - 13 x - 2 = 5 (1 - x - x - 1 \cdot - 1 x^{2}) - 6 (x^{2} - 3 x - 7)$

Step 3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$- x^{2} - 13 x - 2 = 5 (1 - x - x + 1 x^{2}) - 6 (x^{2} - 3 x - 7)$

Step 3.1.3.1.7

Multiply $x^{2}$ by $1$ .

$- x^{2} - 13 x - 2 = 5 (1 - x - x + x^{2}) - 6 (x^{2} - 3 x - 7)$

Step 3.1.3.2

Subtract $x$ from $- x$ .

$- x^{2} - 13 x - 2 = 5 (1 - 2 x + x^{2}) - 6 (x^{2} - 3 x - 7)$

Step 3.1.4

Apply the distributive property.

$- x^{2} - 13 x - 2 = 5 \cdot 1 + 5 (- 2 x) + 5 x^{2} - 6 (x^{2} - 3 x - 7)$

Step 3.1.5

Simplify.

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Step 3.1.5.1

Multiply $5$ by $1$ .

$- x^{2} - 13 x - 2 = 5 + 5 (- 2 x) + 5 x^{2} - 6 (x^{2} - 3 x - 7)$

Step 3.1.5.2

Multiply $- 2$ by $5$ .

$- x^{2} - 13 x - 2 = 5 - 10 x + 5 x^{2} - 6 (x^{2} - 3 x - 7)$

Step 3.1.6

Apply the distributive property.

$- x^{2} - 13 x - 2 = 5 - 10 x + 5 x^{2} - 6 x^{2} - 6 (- 3 x) - 6 \cdot - 7$

Step 3.1.7

Simplify.

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Step 3.1.7.1

Multiply $- 3$ by $- 6$ .

$- x^{2} - 13 x - 2 = 5 - 10 x + 5 x^{2} - 6 x^{2} + 18 x - 6 \cdot - 7$

Step 3.1.7.2

Multiply $- 6$ by $- 7$ .

$- x^{2} - 13 x - 2 = 5 - 10 x + 5 x^{2} - 6 x^{2} + 18 x + 42$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Add $5$ and $42$ .

$- x^{2} - 13 x - 2 = - 10 x + 5 x^{2} - 6 x^{2} + 18 x + 47$

Step 3.2.2

Add $- 10 x$ and $18 x$ .

$- x^{2} - 13 x - 2 = 5 x^{2} - 6 x^{2} + 8 x + 47$

Step 3.2.3

Subtract $6 x^{2}$ from $5 x^{2}$ .

$- x^{2} - 13 x - 2 = - x^{2} + 8 x + 47$

Step 4

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

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Step 4.1

Add $x^{2}$ to both sides of the equation.

$- x^{2} - 13 x - 2 + x^{2} = 8 x + 47$

Step 4.2

Subtract $8 x$ from both sides of the equation.

$- x^{2} - 13 x - 2 + x^{2} - 8 x = 47$

Step 4.3

Combine the opposite terms in $- x^{2} - 13 x - 2 + x^{2} - 8 x$ .

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Step 4.3.1

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

$- 13 x - 2 + 0 - 8 x = 47$

Step 4.3.2

Add $- 13 x - 2$ and $0$ .

$- 13 x - 2 - 8 x = 47$

Step 4.4

Subtract $8 x$ from $- 13 x$ .

$- 21 x - 2 = 47$

Step 5

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

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Step 5.1

Add $2$ to both sides of the equation.

$- 21 x = 47 + 2$

Step 5.2

Add $47$ and $2$ .

$- 21 x = 49$

Step 6

Divide each term in $- 21 x = 49$ by $- 21$ and simplify.

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Step 6.1

Divide each term in $- 21 x = 49$ by $- 21$ .

$\frac{- 21 x}{- 21} = \frac{49}{- 21}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 21$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 21 x}{- 21} = \frac{49}{- 21}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{49}{- 21}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of $49$ and $- 21$ .

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Step 6.3.1.1

Factor $7$ out of $49$ .

$x = \frac{7 (7)}{- 21}$

Step 6.3.1.2

Cancel the common factors.

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Step 6.3.1.2.1

Factor $7$ out of $- 21$ .

$x = \frac{7 \cdot 7}{7 \cdot - 3}$

Step 6.3.1.2.2

Cancel the common factor.

$x = \frac{7 \cdot 7}{7 \cdot - 3}$

Step 6.3.1.2.3

Rewrite the expression.

$x = \frac{7}{- 3}$

Step 6.3.2

Move the negative in front of the fraction.

$x = - \frac{7}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7}{3}$

Decimal Form:

$x = - 2.\overline{3}$

Mixed Number Form:

$x = - 2 \frac{1}{3}$