Finite Math Examples

$2 x^{3} - 19 x^{2} = - 9 x$

Step 1

Add $9 x$ to both sides of the equation.

$2 x^{3} - 19 x^{2} + 9 x = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $x$ out of $2 x^{3} - 19 x^{2} + 9 x$ .

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Step 2.1.1

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

$x (2 x^{2}) - 19 x^{2} + 9 x = 0$

Step 2.1.2

Factor $x$ out of $- 19 x^{2}$ .

$x (2 x^{2}) + x (- 19 x) + 9 x = 0$

Step 2.1.3

Factor $x$ out of $9 x$ .

$x (2 x^{2}) + x (- 19 x) + x \cdot 9 = 0$

Step 2.1.4

Factor $x$ out of $x (2 x^{2}) + x (- 19 x)$ .

$x (2 x^{2} - 19 x) + x \cdot 9 = 0$

Step 2.1.5

Factor $x$ out of $x (2 x^{2} - 19 x) + x \cdot 9$ .

$x (2 x^{2} - 19 x + 9) = 0$

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 $a \cdot c = 2 \cdot 9 = 18$ and whose sum is $b = - 19$ .

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Step 2.2.1.1.1

Factor $- 19$ out of $- 19 x$ .

$x (2 x^{2} - 19 x + 9) = 0$

Step 2.2.1.1.2

Rewrite $- 19$ as $- 1$ plus $- 18$

$x (2 x^{2} + (- 1 - 18) x + 9) = 0$

Step 2.2.1.1.3

Apply the distributive property.

$x (2 x^{2} - 1 x - 18 x + 9) = 0$

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.

$x ((2 x^{2} - 1 x) - 18 x + 9) = 0$

Step 2.2.1.2.2

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

$x (x (2 x - 1) - 9 (2 x - 1)) = 0$

Step 2.2.1.3

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

$x ((2 x - 1) (x - 9)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$x (2 x - 1) (x - 9) = 0$

Step 3

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

$x = 0$

$2 x - 1 = 0$

$x - 9 = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

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

Add $1$ to 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 6

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 6.2

Add $9$ to both sides of the equation.

$x = 9$

Step 7

The final solution is all the values that make $x (2 x - 1) (x - 9) = 0$ true.

$x = 0, \frac{1}{2}, 9$