Pre-Algebra Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Apply the distributive property.

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

Step 1.1.3

Apply the distributive property.

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

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.1.2

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

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Step 1.2.1.2.1

Move $x$ .

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

Step 1.2.1.2.2

Multiply $x$ by $x$ .

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

Step 1.2.1.3

Move $3$ to the left of $x$ .

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

Step 1.2.1.4

Multiply $2$ by $- 1$ .

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

Step 1.2.1.5

Multiply $- 1$ by $3$ .

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

Step 1.2.2

Subtract $2 x$ from $3 x$ .

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

Step 2

Subtract $3$ from both sides of the equation.

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

Step 3

Subtract $3$ from $- 3$ .

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

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = 1$ .

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Step 4.1.1

Multiply by $1$ .

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

Step 4.1.2

Rewrite $1$ as $- 3$ plus $4$

$2 x^{2} + (- 3 + 4) x - 6 = 0$

Step 4.1.3

Apply the distributive property.

$2 x^{2} - 3 x + 4 x - 6 = 0$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$(2 x^{2} - 3 x) + 4 x - 6 = 0$

Step 4.2.2

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

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

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

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

Step 5

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

$2 x - 3 = 0$

$x + 2 = 0$

Step 6

Set $2 x - 3$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Step 6.2

Solve $2 x - 3 = 0$ for $x$ .

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Step 6.2.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 6.2.2

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 6.2.2.1

Divide each term in $2 x = 3$ by $2$ .

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

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 7

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

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Step 7.1

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

$x + 2 = 0$

Step 7.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 8

The final solution is all the values that make $(2 x - 3) (x + 2) = 0$ true.

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