Finite Math Examples

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

Step 1

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

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Step 1.1

Rewrite.

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

Step 1.2

Simplify by adding zeros.

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

Step 1.3

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

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

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 $x$ .

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

Step 1.4.1.2

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

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

Step 1.4.1.3

Multiply $- 3$ by $2$ .

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

Step 1.4.2

Subtract $3 x$ from $2 x$ .

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

Step 2

Simplify $x^{2} \cdot 2 \cdot - 3$ .

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Step 2.1

Move $2$ to the left of $x^{2}$ .

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

Step 2.2

Multiply $- 3$ by $2$ .

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

Step 3

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

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Step 3.1

Add $6 x^{2}$ to both sides of the equation.

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

Step 3.2

Add $x^{2}$ and $6 x^{2}$ .

$7 x^{2} - x - 6 = 0$

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 $a \cdot c = 7 \cdot - 6 = - 42$ and whose sum is $b = - 1$ .

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Step 4.1.1

Factor $- 1$ out of $- x$ .

$7 x^{2} - (x) - 6 = 0$

Step 4.1.2

Rewrite $- 1$ as $6$ plus $- 7$

$7 x^{2} + (6 - 7) x - 6 = 0$

Step 4.1.3

Apply the distributive property.

$7 x^{2} + 6 x - 7 x - 6 = 0$

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.

$(7 x^{2} + 6 x) - 7 x - 6 = 0$

Step 4.2.2

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

$x (7 x + 6) - (7 x + 6) = 0$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $7 x + 6$ .

$(7 x + 6) (x - 1) = 0$

Step 5

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

$7 x + 6 = 0$

$x - 1 = 0$

Step 6

Set $7 x + 6$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $7 x + 6$ equal to $0$ .

$7 x + 6 = 0$

Step 6.2

Solve $7 x + 6 = 0$ for $x$ .

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Step 6.2.1

Subtract $6$ from both sides of the equation.

$7 x = - 6$

Step 6.2.2

Divide each term in $7 x = - 6$ by $7$ and simplify.

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Step 6.2.2.1

Divide each term in $7 x = - 6$ by $7$ .

$\frac{7 x}{7} = \frac{- 6}{7}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $7$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 6}{7}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{7}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{6}{7}$

Step 7

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 7.2

Add $1$ to both sides of the equation.

$x = 1$

Step 8

The final solution is all the values that make $(7 x + 6) (x - 1) = 0$ true.

$x = - \frac{6}{7}, 1$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{6}{7}, 1$

Decimal Form:

$x = - 0.\overline{857142}, 1$