Solve by Factoring 2x^4+16x^3+32x^2=0

$2 x^{4} + 16 x^{3} + 32 x^{2} = 0$

Step 1

Factor $2 x^{2}$ out of $2 x^{4} + 16 x^{3} + 32 x^{2}$ .

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

$2 x^{2} (x^{2}) + 16 x^{3} + 32 x^{2} = 0$

Factor $2 x^{2}$ out of $16 x^{3}$ .

$2 x^{2} (x^{2}) + 2 x^{2} (8 x) + 32 x^{2} = 0$

Factor $2 x^{2}$ out of $32 x^{2}$ .

$2 x^{2} (x^{2}) + 2 x^{2} (8 x) + 2 x^{2} (16) = 0$

Factor $2 x^{2}$ out of $2 x^{2} (x^{2}) + 2 x^{2} (8 x)$ .

$2 x^{2} (x^{2} + 8 x) + 2 x^{2} (16) = 0$

Factor $2 x^{2}$ out of $2 x^{2} (x^{2} + 8 x) + 2 x^{2} (16)$ .

$2 x^{2} (x^{2} + 8 x + 16) = 0$

Step 2

Factor using the perfect square rule.

Rewrite $16$ as $4^{2}$ .

$2 x^{2} (x^{2} + 8 x + 4^{2}) = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

$2 x^{2} (x^{2} + 2 \cdot x \cdot 4 + 4^{2}) = 0$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

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

Step 3

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

$x^{2} = 0$

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

Step 4

Set $x^{2}$ equal to $0$ and solve for $x$ .

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Solve $x^{2} = 0$ for $x$ .

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

Set ${(x + 4)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x + 4)}^{2}$ equal to $0$ .

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

Solve ${(x + 4)}^{2} = 0$ for $x$ .

Set the $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 6

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

$x = 0, - 4$

Solve by Factoring 2x^4+16x^3+32x^2=0

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