Algebra Examples

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

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

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

Step 2

Solve the equation for $x$ .

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Step 2.1

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

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Step 2.1.1

Rewrite.

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

Step 2.1.2

Simplify by adding zeros.

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

Step 2.1.3

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

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Step 2.1.3.1

Apply the distributive property.

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

Step 2.1.3.2

Apply the distributive property.

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

Step 2.1.3.3

Apply the distributive property.

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

Step 2.1.4

Simplify and combine like terms.

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Step 2.1.4.1

Simplify each term.

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Step 2.1.4.1.1

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

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Step 2.1.4.1.1.1

Move $x$ .

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

Step 2.1.4.1.1.2

Multiply $x$ by $x$ .

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

Step 2.1.4.1.2

Multiply $- 1$ by $2$ .

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

Step 2.1.4.1.3

Multiply $- 3$ by $- 1$ .

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

Step 2.1.4.2

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

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

Step 2.2

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

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Step 2.2.1

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

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $x$ by $x$ .

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

Step 2.2.2.1.2

Move $- 2$ to the left of $x$ .

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

Step 2.2.2.1.3

Multiply $- 3$ by $- 2$ .

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

Step 2.2.2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Combine the opposite terms in $2 x^{2} - 5 x + 3 - x^{2} + 5 x$ .

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Step 2.3.3.1

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

$2 x^{2} + 3 - x^{2} + 0 = 6$

Step 2.3.3.2

Add $2 x^{2} + 3 - x^{2}$ and $0$ .

$2 x^{2} + 3 - x^{2} = 6$

Step 2.3.4

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

$x^{2} + 3 = 6$

Step 2.4

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

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Step 2.4.1

Subtract $3$ from both sides of the equation.

$x^{2} = 6 - 3$

Step 2.4.2

Subtract $3$ from $6$ .

$x^{2} = 3$

Step 2.5

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

$x = \pm \sqrt{3}$

Step 2.6

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

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Step 2.6.1

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

$x = \sqrt{3}$

Step 2.6.2

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

$x = - \sqrt{3}$

Step 2.6.3

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

$x = \sqrt{3}, - \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{3}, - \sqrt{3}$

Decimal Form:

$x = 1.73205080 \dots, - 1.73205080 \dots$