Solve for x (1-2x)(1-3x)=(6x-1)x-1

$(1 - 2 x) (1 - 3 x) = (6 x - 1) x - 1$

Step 1

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

Rewrite.

$0 + 0 + (1 - 2 x) (1 - 3 x) = (6 x - 1) x - 1$

Simplify by adding zeros.

$(1 - 2 x) (1 - 3 x) = (6 x - 1) x - 1$

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

Apply the distributive property.

$1 (1 - 3 x) - 2 x (1 - 3 x) = (6 x - 1) x - 1$

Apply the distributive property.

$1 \cdot 1 + 1 (- 3 x) - 2 x (1 - 3 x) = (6 x - 1) x - 1$

Apply the distributive property.

$1 \cdot 1 + 1 (- 3 x) - 2 x \cdot 1 - 2 x (- 3 x) = (6 x - 1) x - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$1 + 1 (- 3 x) - 2 x \cdot 1 - 2 x (- 3 x) = (6 x - 1) x - 1$

Multiply $- 3 x$ by $1$ .

$1 - 3 x - 2 x \cdot 1 - 2 x (- 3 x) = (6 x - 1) x - 1$

Multiply $- 2$ by $1$ .

$1 - 3 x - 2 x - 2 x (- 3 x) = (6 x - 1) x - 1$

Rewrite using the commutative property of multiplication.

$1 - 3 x - 2 x - 2 \cdot - 3 x \cdot x = (6 x - 1) x - 1$

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

Move $x$ .

$1 - 3 x - 2 x - 2 \cdot - 3 (x \cdot x) = (6 x - 1) x - 1$

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 3$ .

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

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

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

Step 2

Simplify each term.

Apply the distributive property.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

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

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

Step 3

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

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

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

Add $x$ to both sides of the equation.

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

Combine the opposite terms in $1 - 5 x + 6 x^{2} - 6 x^{2} + x$ .

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

$1 - 5 x + 0 + x = - 1$

Add $1 - 5 x$ and $0$ .

$1 - 5 x + x = - 1$

Add $- 5 x$ and $x$ .

$1 - 4 x = - 1$

Step 4

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

Subtract $1$ from both sides of the equation.

$- 4 x = - 1 - 1$

Subtract $1$ from $- 1$ .

$- 4 x = - 2$

Step 5

Divide each term in $- 4 x = - 2$ by $- 4$ and simplify.

Divide each term in $- 4 x = - 2$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 2}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 2}{- 4}$

Divide $x$ by $1$ .

$x = \frac{- 2}{- 4}$

Simplify the right side.

Cancel the common factor of $- 2$ and $- 4$ .

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

$x = \frac{- 2 (1)}{- 4}$

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.5$

Solve for x (1-2x)(1-3x)=(6x-1)x-1

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