Solve by Factoring 14x^2-24x-2x^3=0

$14 x^{2} - 24 x - 2 x^{3} = 0$

Step 1

Factor $2 x$ out of $14 x^{2} - 24 x - 2 x^{3}$ .

Factor $2 x$ out of $14 x^{2}$ .

$2 x (7 x) - 24 x - 2 x^{3} = 0$

Factor $2 x$ out of $- 24 x$ .

$2 x (7 x) + 2 x (- 12) - 2 x^{3} = 0$

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

$2 x (7 x) + 2 x (- 12) + 2 x (- x^{2}) = 0$

Factor $2 x$ out of $2 x (7 x) + 2 x (- 12)$ .

$2 x (7 x - 12) + 2 x (- x^{2}) = 0$

Factor $2 x$ out of $2 x (7 x - 12) + 2 x (- x^{2})$ .

$2 x (7 x - 12 - x^{2}) = 0$

Step 2

Factor.

Factor by grouping.

Reorder terms.

$2 x (- x^{2} + 7 x - 12) = 0$

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 $a \cdot c = - 1 \cdot - 12 = 12$ and whose sum is $b = 7$ .

Factor $7$ out of $7 x$ .

$2 x (- x^{2} + 7 (x) - 12) = 0$

Rewrite $7$ as $3$ plus $4$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Remove unnecessary parentheses.

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

Step 3

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

$x = 0$

$- x + 3 = 0$

$x - 4 = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

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

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

$- x + 3 = 0$

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

Subtract $3$ from both sides of the equation.

$- x = - 3$

Divide each term in $- x = - 3$ by $- 1$ and simplify.

Divide each term in $- x = - 3$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 3$ by $- 1$ .

$x = 3$

Step 6

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

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

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Step 7

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

$x = 0, 3, 4$

Solve by Factoring 14x^2-24x-2x^3=0

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