Solve by Factoring 3x^3+49=x^2+147x

$3 x^{3} + 49 = x^{2} + 147 x$

Step 1

Move all the expressions to the left side of the equation.

Subtract $x^{2}$ from both sides of the equation.

$3 x^{3} + 49 - x^{2} = 147 x$

Subtract $147 x$ from both sides of the equation.

$3 x^{3} + 49 - x^{2} - 147 x = 0$

Step 2

Reorder terms.

$3 x^{3} - x^{2} - 147 x + 49 = 0$

Step 3

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{3} - x^{2}) - 147 x + 49 = 0$

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

$x^{2} (3 x - 1) - 49 (3 x - 1) = 0$

Step 4

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(3 x - 1) (x^{2} - 49) = 0$

Step 5

Rewrite $49$ as $7^{2}$ .

$(3 x - 1) (x^{2} - 7^{2}) = 0$

Step 6

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 7$ .

$(3 x - 1) ((x + 7) (x - 7)) = 0$

Remove unnecessary parentheses.

$(3 x - 1) (x + 7) (x - 7) = 0$

Step 7

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

$3 x - 1 = 0$

$x + 7 = 0$

$x - 7 = 0$

Step 8

Set $3 x - 1$ equal to $0$ and solve for $x$ .

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Solve $3 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$3 x = 1$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 9

Set $x + 7$ equal to $0$ and solve for $x$ .

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 10

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

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

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Step 11

The final solution is all the values that make $(3 x - 1) (x + 7) (x - 7) = 0$ true.

$x = \frac{1}{3}, - 7, 7$

Solve by Factoring 3x^3+49=x^2+147x

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