Algebra Examples

$3 x \cdot (2 x - 1) - 12 \cdot (2 x - 1) = 0$

Step 1

Simplify $3 x \cdot (2 x - 1) - 12 \cdot (2 x - 1)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$3 x (2 x) + 3 x \cdot - 1 - 12 \cdot (2 x - 1) = 0$

Step 1.1.2

Rewrite using the commutative property of multiplication.

$3 \cdot 2 x \cdot x + 3 x \cdot - 1 - 12 \cdot (2 x - 1) = 0$

Step 1.1.3

Multiply $- 1$ by $3$ .

$3 \cdot 2 x \cdot x - 3 x - 12 \cdot (2 x - 1) = 0$

Step 1.1.4

Simplify each term.

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Step 1.1.4.1

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

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Step 1.1.4.1.1

Move $x$ .

$3 \cdot 2 (x \cdot x) - 3 x - 12 \cdot (2 x - 1) = 0$

Step 1.1.4.1.2

Multiply $x$ by $x$ .

$3 \cdot 2 x^{2} - 3 x - 12 \cdot (2 x - 1) = 0$

Step 1.1.4.2

Multiply $3$ by $2$ .

$6 x^{2} - 3 x - 12 \cdot (2 x - 1) = 0$

Step 1.1.5

Apply the distributive property.

$6 x^{2} - 3 x - 12 (2 x) - 12 \cdot - 1 = 0$

Step 1.1.6

Multiply $2$ by $- 12$ .

$6 x^{2} - 3 x - 24 x - 12 \cdot - 1 = 0$

Step 1.1.7

Multiply $- 12$ by $- 1$ .

$6 x^{2} - 3 x - 24 x + 12 = 0$

Step 1.2

Subtract $24 x$ from $- 3 x$ .

$6 x^{2} - 27 x + 12 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $3$ out of $6 x^{2} - 27 x + 12$ .

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Step 2.1.1

Factor $3$ out of $6 x^{2}$ .

$3 (2 x^{2}) - 27 x + 12 = 0$

Step 2.1.2

Factor $3$ out of $- 27 x$ .

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

Step 2.1.3

Factor $3$ out of $12$ .

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

Step 2.1.4

Factor $3$ out of $3 (2 x^{2}) + 3 (- 9 x)$ .

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

Step 2.1.5

Factor $3$ out of $3 (2 x^{2} - 9 x) + 3 (4)$ .

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

Step 2.2

Factor.

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Step 2.2.1

Factor by grouping.

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Step 2.2.1.1

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 $a \cdot c = 2 \cdot 4 = 8$ and whose sum is $b = - 9$ .

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Step 2.2.1.1.1

Factor $- 9$ out of $- 9 x$ .

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

Step 2.2.1.1.2

Rewrite $- 9$ as $- 1$ plus $- 8$

$3 (2 x^{2} + (- 1 - 8) x + 4) = 0$

Step 2.2.1.1.3

Apply the distributive property.

$3 (2 x^{2} - 1 x - 8 x + 4) = 0$

Step 2.2.1.2

Factor out the greatest common factor from each group.

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Step 2.2.1.2.1

Group the first two terms and the last two terms.

$3 ((2 x^{2} - 1 x) - 8 x + 4) = 0$

Step 2.2.1.2.2

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

$3 (x (2 x - 1) - 4 (2 x - 1)) = 0$

Step 2.2.1.3

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

$3 ((2 x - 1) (x - 4)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$3 (2 x - 1) (x - 4) = 0$

Step 3

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

$2 x - 1 = 0$

$x - 4 = 0$

Step 4

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

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Step 4.1

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

$2 x - 1 = 0$

Step 4.2

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

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Step 4.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 4.2.2

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

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Step 4.2.2.1

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

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

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 5

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

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Step 5.1

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

$x - 4 = 0$

Step 5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 6

The final solution is all the values that make $3 (2 x - 1) (x - 4) = 0$ true.

$x = \frac{1}{2}, 4$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}, 4$

Decimal Form:

$x = 0.5, 4$