Algebra Examples

$(1 - 2 x) (1 - 3 x) = (5 x - 1) x - 2 (2 x - 5)$

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.

$(5 x - 1) x - 2 (2 x - 5) = (1 - 2 x) (1 - 3 x)$

Step 2

Simplify $(5 x - 1) x - 2 (2 x - 5)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$5 x \cdot x - 1 x - 2 (2 x - 5) = (1 - 2 x) (1 - 3 x)$

Step 2.1.2

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

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Step 2.1.2.1

Move $x$ .

$5 (x \cdot x) - 1 x - 2 (2 x - 5) = (1 - 2 x) (1 - 3 x)$

Step 2.1.2.2

Multiply $x$ by $x$ .

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

Step 2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.1.4

Apply the distributive property.

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

Step 2.1.5

Multiply $2$ by $- 2$ .

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

Step 2.1.6

Multiply $- 2$ by $- 5$ .

$5 x^{2} - x - 4 x + 10 = (1 - 2 x) (1 - 3 x)$

Step 2.2

Subtract $4 x$ from $- x$ .

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

Step 3

Simplify $(1 - 2 x) (1 - 3 x)$ .

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Step 3.1

Expand $(1 - 2 x) (1 - 3 x)$ using the FOIL Method.

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Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

Apply the distributive property.

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

Step 3.1.3

Apply the distributive property.

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

Step 3.2

Simplify and combine like terms.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $1$ by $1$ .

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

Step 3.2.1.2

Multiply $- 3 x$ by $1$ .

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

Step 3.2.1.3

Multiply $- 2$ by $1$ .

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

Step 3.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.5

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

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Step 3.2.1.5.1

Move $x$ .

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

Step 3.2.1.5.2

Multiply $x$ by $x$ .

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

Step 3.2.1.6

Multiply $- 2$ by $- 3$ .

$5 x^{2} - 5 x + 10 = 1 - 3 x - 2 x + 6 x^{2}$

Step 3.2.2

Subtract $2 x$ from $- 3 x$ .

$5 x^{2} - 5 x + 10 = 1 - 5 x + 6 x^{2}$

Step 4

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

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Step 4.1

Add $5 x$ to both sides of the equation.

$5 x^{2} - 5 x + 10 + 5 x = 1 + 6 x^{2}$

Step 4.2

Subtract $6 x^{2}$ from both sides of the equation.

$5 x^{2} - 5 x + 10 + 5 x - 6 x^{2} = 1$

Step 4.3

Combine the opposite terms in $5 x^{2} - 5 x + 10 + 5 x - 6 x^{2}$ .

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Step 4.3.1

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

$5 x^{2} + 0 + 10 - 6 x^{2} = 1$

Step 4.3.2

Add $5 x^{2}$ and $0$ .

$5 x^{2} + 10 - 6 x^{2} = 1$

Step 4.4

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

$- x^{2} + 10 = 1$

Step 5

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

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Step 5.1

Subtract $10$ from both sides of the equation.

$- x^{2} = 1 - 10$

Step 5.2

Subtract $10$ from $1$ .

$- x^{2} = - 9$

Step 6

Divide each term in $- x^{2} = - 9$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- x^{2} = - 9$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 9}{- 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 9}{- 1}$

Step 6.2.2

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

$x^{2} = \frac{- 9}{- 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $- 9$ by $- 1$ .

$x^{2} = 9$

Step 7

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{9}$

Step 8

Simplify $\pm \sqrt{9}$ .

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Step 8.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Step 8.2

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

$x = \pm 3$

Step 9

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

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Step 9.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

Step 9.2

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

$x = - 3$

Step 9.3

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

$x = 3, - 3$