Finite Math Examples

$20 - 3 (- 6 x^{2} + 22 x - 16) = 12 x - 4$

Step 1

Simplify $20 - 3 (- 6 x^{2} + 22 x - 16)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$20 - 3 (- 6 x^{2}) - 3 (22 x) - 3 \cdot - 16 = 12 x - 4$

Step 1.1.2

Simplify.

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Step 1.1.2.1

Multiply $- 6$ by $- 3$ .

$20 + 18 x^{2} - 3 (22 x) - 3 \cdot - 16 = 12 x - 4$

Step 1.1.2.2

Multiply $22$ by $- 3$ .

$20 + 18 x^{2} - 66 x - 3 \cdot - 16 = 12 x - 4$

Step 1.1.2.3

Multiply $- 3$ by $- 16$ .

$20 + 18 x^{2} - 66 x + 48 = 12 x - 4$

Step 1.2

Add $20$ and $48$ .

$18 x^{2} - 66 x + 68 = 12 x - 4$

Step 2

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

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Step 2.1

Subtract $12 x$ from both sides of the equation.

$18 x^{2} - 66 x + 68 - 12 x = - 4$

Step 2.2

Subtract $12 x$ from $- 66 x$ .

$18 x^{2} - 78 x + 68 = - 4$

Step 3

Add $4$ to both sides of the equation.

$18 x^{2} - 78 x + 68 + 4 = 0$

Step 4

Add $68$ and $4$ .

$18 x^{2} - 78 x + 72 = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Factor $6$ out of $18 x^{2} - 78 x + 72$ .

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Step 5.1.1

Factor $6$ out of $18 x^{2}$ .

$6 (3 x^{2}) - 78 x + 72 = 0$

Step 5.1.2

Factor $6$ out of $- 78 x$ .

$6 (3 x^{2}) + 6 (- 13 x) + 72 = 0$

Step 5.1.3

Factor $6$ out of $72$ .

$6 (3 x^{2}) + 6 (- 13 x) + 6 (12) = 0$

Step 5.1.4

Factor $6$ out of $6 (3 x^{2}) + 6 (- 13 x)$ .

$6 (3 x^{2} - 13 x) + 6 (12) = 0$

Step 5.1.5

Factor $6$ out of $6 (3 x^{2} - 13 x) + 6 (12)$ .

$6 (3 x^{2} - 13 x + 12) = 0$

Step 5.2

Factor.

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Step 5.2.1

Factor by grouping.

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Step 5.2.1.1

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 $a \cdot c = 3 \cdot 12 = 36$ and whose sum is $b = - 13$ .

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Step 5.2.1.1.1

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

$6 (3 x^{2} - 13 x + 12) = 0$

Step 5.2.1.1.2

Rewrite $- 13$ as $- 4$ plus $- 9$

$6 (3 x^{2} + (- 4 - 9) x + 12) = 0$

Step 5.2.1.1.3

Apply the distributive property.

$6 (3 x^{2} - 4 x - 9 x + 12) = 0$

Step 5.2.1.2

Factor out the greatest common factor from each group.

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Step 5.2.1.2.1

Group the first two terms and the last two terms.

$6 ((3 x^{2} - 4 x) - 9 x + 12) = 0$

Step 5.2.1.2.2

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

$6 (x (3 x - 4) - 3 (3 x - 4)) = 0$

Step 5.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 4$ .

$6 ((3 x - 4) (x - 3)) = 0$

Step 5.2.2

Remove unnecessary parentheses.

$6 (3 x - 4) (x - 3) = 0$

Step 6

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

$3 x - 4 = 0$

$x - 3 = 0$

Step 7

Set $3 x - 4$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $3 x - 4$ equal to $0$ .

$3 x - 4 = 0$

Step 7.2

Solve $3 x - 4 = 0$ for $x$ .

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Step 7.2.1

Add $4$ to both sides of the equation.

$3 x = 4$

Step 7.2.2

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 7.2.2.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $3$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 8

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

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Step 8.1

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

$x - 3 = 0$

Step 8.2

Add $3$ to both sides of the equation.

$x = 3$

Step 9

The final solution is all the values that make $6 (3 x - 4) (x - 3) = 0$ true.

$x = \frac{4}{3}, 3$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{3}, 3$

Decimal Form:

$x = 1.\overline{3}, 3$

Mixed Number Form:

$x = 1 \frac{1}{3}, 3$