Solve by Factoring 4x^7-28x^6=-48x^5

$4 x^{7} - 28 x^{6} = - 48 x^{5}$

Step 1

Add $48 x^{5}$ to both sides of the equation.

$4 x^{7} - 28 x^{6} + 48 x^{5} = 0$

Step 2

Factor $4 x^{5}$ out of $4 x^{7} - 28 x^{6} + 48 x^{5}$ .

Factor $4 x^{5}$ out of $4 x^{7}$ .

$4 x^{5} (x^{2}) - 28 x^{6} + 48 x^{5} = 0$

Factor $4 x^{5}$ out of $- 28 x^{6}$ .

$4 x^{5} (x^{2}) + 4 x^{5} (- 7 x) + 48 x^{5} = 0$

Factor $4 x^{5}$ out of $48 x^{5}$ .

$4 x^{5} (x^{2}) + 4 x^{5} (- 7 x) + 4 x^{5} (12) = 0$

Factor $4 x^{5}$ out of $4 x^{5} (x^{2}) + 4 x^{5} (- 7 x)$ .

$4 x^{5} (x^{2} - 7 x) + 4 x^{5} (12) = 0$

Factor $4 x^{5}$ out of $4 x^{5} (x^{2} - 7 x) + 4 x^{5} (12)$ .

$4 x^{5} (x^{2} - 7 x + 12) = 0$

Step 3

Factor.

Factor $x^{2} - 7 x + 12$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $- 7$ .

$- 4, - 3$

Write the factored form using these integers.

$4 x^{5} ((x - 4) (x - 3)) = 0$

Remove unnecessary parentheses.

$4 x^{5} (x - 4) (x - 3) = 0$

Step 4

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

$x^{5} = 0$

$x - 4 = 0$

$x - 3 = 0$

Step 5

Set $x^{5}$ equal to $0$ and solve for $x$ .

Set $x^{5}$ equal to $0$ .

$x^{5} = 0$

Solve $x^{5} = 0$ for $x$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[5]{0}$

Simplify $\sqrt[5]{0}$ .

Rewrite $0$ as $0^{5}$ .

$x = \sqrt[5]{0^{5}}$

Pull terms out from under the radical, assuming real numbers.

$x = 0$

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

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 8

The final solution is all the values that make $4 x^{5} (x - 4) (x - 3) = 0$ true.

$x = 0, 4, 3$

Solve by Factoring 4x^7-28x^6=-48x^5

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