Pre-Algebra Examples

$(2 x + 7) (x - 1) = 18 x$

Step 1

Simplify $(2 x + 7) (x - 1)$ .

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Step 1.1

Rewrite.

$0 + 0 + (2 x + 7) (x - 1) = 18 x$

Step 1.2

Simplify by adding zeros.

$(2 x + 7) (x - 1) = 18 x$

Step 1.3

Expand $(2 x + 7) (x - 1)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$2 x (x - 1) + 7 (x - 1) = 18 x$

Step 1.3.2

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 1 + 7 (x - 1) = 18 x$

Step 1.3.3

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 1 + 7 x + 7 \cdot - 1 = 18 x$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

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

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Step 1.4.1.1.1

Move $x$ .

$2 (x \cdot x) + 2 x \cdot - 1 + 7 x + 7 \cdot - 1 = 18 x$

Step 1.4.1.1.2

Multiply $x$ by $x$ .

$2 x^{2} + 2 x \cdot - 1 + 7 x + 7 \cdot - 1 = 18 x$

Step 1.4.1.2

Multiply $- 1$ by $2$ .

$2 x^{2} - 2 x + 7 x + 7 \cdot - 1 = 18 x$

Step 1.4.1.3

Multiply $7$ by $- 1$ .

$2 x^{2} - 2 x + 7 x - 7 = 18 x$

Step 1.4.2

Add $- 2 x$ and $7 x$ .

$2 x^{2} + 5 x - 7 = 18 x$

Step 2

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

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Step 2.1

Subtract $18 x$ from both sides of the equation.

$2 x^{2} + 5 x - 7 - 18 x = 0$

Step 2.2

Subtract $18 x$ from $5 x$ .

$2 x^{2} - 13 x - 7 = 0$

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 7 = - 14$ and whose sum is $b = - 13$ .

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Step 3.1.1

Factor $- 13$ out of $- 13 x$ .

$2 x^{2} - 13 x - 7 = 0$

Step 3.1.2

Rewrite $- 13$ as $1$ plus $- 14$

$2 x^{2} + (1 - 14) x - 7 = 0$

Step 3.1.3

Apply the distributive property.

$2 x^{2} + 1 x - 14 x - 7 = 0$

Step 3.1.4

Multiply $x$ by $1$ .

$2 x^{2} + x - 14 x - 7 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(2 x^{2} + x) - 14 x - 7 = 0$

Step 3.2.2

Factor out the greatest common factor (GCF) from each group.

$x (2 x + 1) - 7 (2 x + 1) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$(2 x + 1) (x - 7) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x + 1 = 0$

$x - 7 = 0$

Step 5

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 5.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 5.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

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

Step 6

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 6.2

Add $7$ to both sides of the equation.

$x = 7$

Step 7

The final solution is all the values that make $(2 x + 1) (x - 7) = 0$ true.

$x = - \frac{1}{2}, 7$