Solve for x 6x^3+21x^2-90x=0

$6 x^{3} + 21 x^{2} - 90 x = 0$

Factor the left side of the equation.

Factor $3 x$ out of $6 x^{3} + 21 x^{2} - 90 x$ .

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

$3 x (2 x^{2}) + 21 x^{2} - 90 x = 0$

Factor $3 x$ out of $21 x^{2}$ .

$3 x (2 x^{2}) + 3 x (7 x) - 90 x = 0$

Factor $3 x$ out of $- 90 x$ .

$3 x (2 x^{2}) + 3 x (7 x) + 3 x (- 30) = 0$

Factor $3 x$ out of $3 x (2 x^{2}) + 3 x (7 x)$ .

$3 x (2 x^{2} + 7 x) + 3 x (- 30) = 0$

Factor $3 x$ out of $3 x (2 x^{2} + 7 x) + 3 x (- 30)$ .

$3 x (2 x^{2} + 7 x - 30) = 0$

Factor.

Factor by grouping.

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 $a \cdot c = 2 \cdot - 30 = - 60$ and whose sum is $b = 7$ .

Factor $7$ out of $7 x$ .

$3 x (2 x^{2} + 7 (x) - 30) = 0$

Rewrite $7$ as $- 5$ plus $12$

$3 x (2 x^{2} + (- 5 + 12) x - 30) = 0$

Apply the distributive property.

$3 x (2 x^{2} - 5 x + 12 x - 30) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$3 x ((2 x^{2} - 5 x) + 12 x - 30) = 0$

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

$3 x (x (2 x - 5) + 6 (2 x - 5)) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x - 5$ .

$3 x ((2 x - 5) (x + 6)) = 0$

Remove unnecessary parentheses.

$3 x (2 x - 5) (x + 6) = 0$

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

$x = 0$

$2 x - 5 = 0$

$x + 6 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $2 x - 5$ equal to $0$ and solve for $x$ .

Set $2 x - 5$ equal to $0$ .

$2 x - 5 = 0$

Solve $2 x - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$2 x = 5$

Divide each term in $2 x = 5$ by $2$ and simplify.

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Divide $x$ by $1$ .

$x = \frac{5}{2}$

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

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

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $3 x (2 x - 5) (x + 6) = 0$ true.

$x = 0, \frac{5}{2}, - 6$

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{5}{2}, - 6$

Decimal Form:

$x = 0, 2.5, - 6$

Mixed Number Form:

$x = 0, 2 \frac{1}{2}, - 6$

Solve for x 6x^3+21x^2-90x=0

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