Algebra Examples

$\frac{2}{3} (x - 4) (x + 5) = 1$

Step 1

Subtract $1$ from both sides of the equation.

$\frac{2}{3} (x - 4) (x + 5) - 1 = 0$

Step 2

Simplify $\frac{2}{3} (x - 4) (x + 5) - 1$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

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

Step 2.1.2

Combine $\frac{2}{3}$ and $x$ .

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

Step 2.1.3

Multiply $\frac{2}{3} \cdot - 4$ .

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Step 2.1.3.1

Combine $\frac{2}{3}$ and $- 4$ .

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

Step 2.1.3.2

Multiply $2$ by $- 4$ .

$(\frac{2 x}{3} + \frac{- 8}{3}) (x + 5) - 1 = 0$

Step 2.1.4

Move the negative in front of the fraction.

$(\frac{2 x}{3} - \frac{8}{3}) (x + 5) - 1 = 0$

Step 2.1.5

Expand $(\frac{2 x}{3} - \frac{8}{3}) (x + 5)$ using the FOIL Method.

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Step 2.1.5.1

Apply the distributive property.

$\frac{2 x}{3} (x + 5) - \frac{8}{3} (x + 5) - 1 = 0$

Step 2.1.5.2

Apply the distributive property.

$\frac{2 x}{3} x + \frac{2 x}{3} \cdot 5 - \frac{8}{3} (x + 5) - 1 = 0$

Step 2.1.5.3

Apply the distributive property.

$\frac{2 x}{3} x + \frac{2 x}{3} \cdot 5 - \frac{8}{3} x - \frac{8}{3} \cdot 5 - 1 = 0$

Step 2.1.6

Simplify and combine like terms.

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Step 2.1.6.1

Simplify each term.

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Step 2.1.6.1.1

Multiply $\frac{2 x}{3} x$ .

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Step 2.1.6.1.1.1

Combine $\frac{2 x}{3}$ and $x$ .

$\frac{2 x \cdot x}{3} + \frac{2 x}{3} \cdot 5 - \frac{8}{3} x - \frac{8}{3} \cdot 5 - 1 = 0$

Step 2.1.6.1.1.2

Raise $x$ to the power of $1$ .

$\frac{2 (x^{1} x)}{3} + \frac{2 x}{3} \cdot 5 - \frac{8}{3} x - \frac{8}{3} \cdot 5 - 1 = 0$

Step 2.1.6.1.1.3

Raise $x$ to the power of $1$ .

$\frac{2 (x^{1} x^{1})}{3} + \frac{2 x}{3} \cdot 5 - \frac{8}{3} x - \frac{8}{3} \cdot 5 - 1 = 0$

Step 2.1.6.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 x^{1 + 1}}{3} + \frac{2 x}{3} \cdot 5 - \frac{8}{3} x - \frac{8}{3} \cdot 5 - 1 = 0$

Step 2.1.6.1.1.5

Add $1$ and $1$ .

$\frac{2 x^{2}}{3} + \frac{2 x}{3} \cdot 5 - \frac{8}{3} x - \frac{8}{3} \cdot 5 - 1 = 0$

Step 2.1.6.1.2

Multiply $\frac{2 x}{3} \cdot 5$ .

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Step 2.1.6.1.2.1

Combine $\frac{2 x}{3}$ and $5$ .

$\frac{2 x^{2}}{3} + \frac{2 x \cdot 5}{3} - \frac{8}{3} x - \frac{8}{3} \cdot 5 - 1 = 0$

Step 2.1.6.1.2.2

Multiply $5$ by $2$ .

$\frac{2 x^{2}}{3} + \frac{10 x}{3} - \frac{8}{3} x - \frac{8}{3} \cdot 5 - 1 = 0$

Step 2.1.6.1.3

Combine $x$ and $\frac{8}{3}$ .

$\frac{2 x^{2}}{3} + \frac{10 x}{3} - \frac{x \cdot 8}{3} - \frac{8}{3} \cdot 5 - 1 = 0$

Step 2.1.6.1.4

Move $8$ to the left of $x$ .

$\frac{2 x^{2}}{3} + \frac{10 x}{3} - \frac{8 \cdot x}{3} - \frac{8}{3} \cdot 5 - 1 = 0$

Step 2.1.6.1.5

Multiply $- \frac{8}{3} \cdot 5$ .

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Step 2.1.6.1.5.1

Multiply $5$ by $- 1$ .

$\frac{2 x^{2}}{3} + \frac{10 x}{3} - \frac{8 x}{3} - 5 (\frac{8}{3}) - 1 = 0$

Step 2.1.6.1.5.2

Combine $- 5$ and $\frac{8}{3}$ .

$\frac{2 x^{2}}{3} + \frac{10 x}{3} - \frac{8 x}{3} + \frac{- 5 \cdot 8}{3} - 1 = 0$

Step 2.1.6.1.5.3

Multiply $- 5$ by $8$ .

$\frac{2 x^{2}}{3} + \frac{10 x}{3} - \frac{8 x}{3} + \frac{- 40}{3} - 1 = 0$

Step 2.1.6.1.6

Move the negative in front of the fraction.

$\frac{2 x^{2}}{3} + \frac{10 x}{3} - \frac{8 x}{3} - \frac{40}{3} - 1 = 0$

Step 2.1.6.2

Combine the numerators over the common denominator.

$\frac{2 x^{2}}{3} + \frac{10 x - 8 x}{3} - \frac{40}{3} - 1 = 0$

Step 2.1.7

Combine the numerators over the common denominator.

$\frac{2 x^{2} + 10 x - 8 x - 40}{3} - 1 = 0$

Step 2.1.8

Subtract $8 x$ from $10 x$ .

$\frac{2 x^{2} + 2 x - 40}{3} - 1 = 0$

Step 2.1.9

Simplify the numerator.

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Step 2.1.9.1

Factor $2$ out of $2 x^{2} + 2 x - 40$ .

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Step 2.1.9.1.1

Factor $2$ out of $2 x^{2}$ .

$\frac{2 (x^{2}) + 2 x - 40}{3} - 1 = 0$

Step 2.1.9.1.2

Factor $2$ out of $2 x$ .

$\frac{2 (x^{2}) + 2 (x) - 40}{3} - 1 = 0$

Step 2.1.9.1.3

Factor $2$ out of $- 40$ .

$\frac{2 (x^{2}) + 2 x + 2 \cdot - 20}{3} - 1 = 0$

Step 2.1.9.1.4

Factor $2$ out of $2 (x^{2}) + 2 x$ .

$\frac{2 (x^{2} + x) + 2 \cdot - 20}{3} - 1 = 0$

Step 2.1.9.1.5

Factor $2$ out of $2 (x^{2} + x) + 2 \cdot - 20$ .

$\frac{2 (x^{2} + x - 20)}{3} - 1 = 0$

Step 2.1.9.2

Factor $x^{2} + x - 20$ using the AC method.

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Step 2.1.9.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 20$ and whose sum is $1$ .

$- 4, 5$

Step 2.1.9.2.2

Write the factored form using these integers.

$\frac{2 ((x - 4) (x + 5))}{3} - 1 = 0$

$\frac{2 (x - 4) (x + 5)}{3} - 1 = 0$

Step 2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.3

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

Multiply $2$ by $- 4$ .

$\frac{(2 x - 8) (x + 5) - 1 \cdot 3}{3} = 0$

Step 2.5.3

Expand $(2 x - 8) (x + 5)$ using the FOIL Method.

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Step 2.5.3.1

Apply the distributive property.

$\frac{2 x (x + 5) - 8 (x + 5) - 1 \cdot 3}{3} = 0$

Step 2.5.3.2

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot 5 - 8 (x + 5) - 1 \cdot 3}{3} = 0$

Step 2.5.3.3

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot 5 - 8 x - 8 \cdot 5 - 1 \cdot 3}{3} = 0$

Step 2.5.4

Simplify and combine like terms.

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Step 2.5.4.1

Simplify each term.

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Step 2.5.4.1.1

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

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Step 2.5.4.1.1.1

Move $x$ .

$\frac{2 (x \cdot x) + 2 x \cdot 5 - 8 x - 8 \cdot 5 - 1 \cdot 3}{3} = 0$

Step 2.5.4.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 x \cdot 5 - 8 x - 8 \cdot 5 - 1 \cdot 3}{3} = 0$

Step 2.5.4.1.2

Multiply $5$ by $2$ .

$\frac{2 x^{2} + 10 x - 8 x - 8 \cdot 5 - 1 \cdot 3}{3} = 0$

Step 2.5.4.1.3

Multiply $- 8$ by $5$ .

$\frac{2 x^{2} + 10 x - 8 x - 40 - 1 \cdot 3}{3} = 0$

Step 2.5.4.2

Subtract $8 x$ from $10 x$ .

$\frac{2 x^{2} + 2 x - 40 - 1 \cdot 3}{3} = 0$

Step 2.5.5

Multiply $- 1$ by $3$ .

$\frac{2 x^{2} + 2 x - 40 - 3}{3} = 0$

Step 2.5.6

Subtract $3$ from $- 40$ .

$\frac{2 x^{2} + 2 x - 43}{3} = 0$

Step 3

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c = 0$

Step 4

Split the fraction $\frac{2 x^{2} + 2 x - 43}{3}$ into two fractions.

$\frac{2 x^{2} + 2 x}{3} + \frac{- 43}{3} = 0$

Step 5

Simplify each term.

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Step 5.1

Factor $2 x$ out of $2 x^{2} + 2 x$ .

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Step 5.1.1

Factor $2 x$ out of $2 x^{2}$ .

$\frac{2 x (x) + 2 x}{3} + \frac{- 43}{3} = 0$

Step 5.1.2

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

$\frac{2 x (x) + 2 x (1)}{3} + \frac{- 43}{3} = 0$

Step 5.1.3

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

$\frac{2 x (x + 1)}{3} + \frac{- 43}{3} = 0$

Step 5.2

Move the negative in front of the fraction.

$\frac{2 x (x + 1)}{3} - \frac{43}{3} = 0$

Step 6

Reorder terms.

$\frac{2}{3} \cdot (x (x + 1)) - \frac{43}{3} = 0$

Step 7