Solve by Factoring 4x^3-26x^2=-42x

$4 x^{3} - 26 x^{2} = - 42 x$

Step 1

Add $42 x$ to both sides of the equation.

$4 x^{3} - 26 x^{2} + 42 x = 0$

Step 2

Factor $2 x$ out of $4 x^{3} - 26 x^{2} + 42 x$ .

Factor $2 x$ out of $4 x^{3}$ .

$2 x (2 x^{2}) - 26 x^{2} + 42 x = 0$

Factor $2 x$ out of $- 26 x^{2}$ .

$2 x (2 x^{2}) + 2 x (- 13 x) + 42 x = 0$

Factor $2 x$ out of $42 x$ .

$2 x (2 x^{2}) + 2 x (- 13 x) + 2 x (21) = 0$

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

$2 x (2 x^{2} - 13 x) + 2 x (21) = 0$

Factor $2 x$ out of $2 x (2 x^{2} - 13 x) + 2 x (21)$ .

$2 x (2 x^{2} - 13 x + 21) = 0$

Step 3

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 $a \cdot c = 2 \cdot 21 = 42$ and whose sum is $b = - 13$ .

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

$2 x (2 x^{2} - 13 x + 21) = 0$

Rewrite $- 13$ as $- 6$ plus $- 7$

$2 x (2 x^{2} + (- 6 - 7) x + 21) = 0$

Apply the distributive property.

$2 x (2 x^{2} - 6 x - 7 x + 21) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$2 x ((2 x^{2} - 6 x) - 7 x + 21) = 0$

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

$2 x (2 x (x - 3) - 7 (x - 3)) = 0$

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

$2 x ((x - 3) (2 x - 7)) = 0$

Remove unnecessary parentheses.

$2 x (x - 3) (2 x - 7) = 0$

Step 4

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

$x = 0$

$x - 3 = 0$

$2 x - 7 = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 7

Set $2 x - 7$ equal to $0$ and solve for $x$ .

Set $2 x - 7$ equal to $0$ .

$2 x - 7 = 0$

Solve $2 x - 7 = 0$ for $x$ .

Add $7$ to both sides of the equation.

$2 x = 7$

Divide each term in $2 x = 7$ by $2$ and simplify.

Divide each term in $2 x = 7$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2}$

Divide $x$ by $1$ .

$x = \frac{7}{2}$

Step 8

The final solution is all the values that make $2 x (x - 3) (2 x - 7) = 0$ true.

$x = 0, 3, \frac{7}{2}$

Solve by Factoring 4x^3-26x^2=-42x

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