Solve by Factoring x^6-63x^3-64=0

$x^{6} - 63 x^{3} - 64 = 0$

Step 1

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

${(x^{3})}^{2} - 63 x^{3} - 64 = 0$

Step 2

Let $u = x^{3}$ . Substitute $u$ for all occurrences of $x^{3}$ .

$u^{2} - 63 u - 64 = 0$

Step 3

Factor $u^{2} - 63 u - 64$ 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 $- 64$ and whose sum is $- 63$ .

$- 64, 1$

Write the factored form using these integers.

$(u - 64) (u + 1) = 0$

Step 4

Replace all occurrences of $u$ with $x^{3}$ .

$(x^{3} - 64) (x^{3} + 1) = 0$

Step 5

Rewrite $64$ as $4^{3}$ .

$(x^{3} - 4^{3}) (x^{3} + 1) = 0$

Step 6

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 4$ .

$(x - 4) (x^{2} + x \cdot 4 + 4^{2}) (x^{3} + 1) = 0$

Step 7

Simplify.

Move $4$ to the left of $x$ .

$(x - 4) (x^{2} + 4 x + 4^{2}) (x^{3} + 1) = 0$

Raise $4$ to the power of $2$ .

$(x - 4) (x^{2} + 4 x + 16) (x^{3} + 1) = 0$

Step 8

Rewrite $1$ as $1^{3}$ .

$(x - 4) (x^{2} + 4 x + 16) (x^{3} + 1^{3}) = 0$

Step 9

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 1$ .

$(x - 4) (x^{2} + 4 x + 16) ((x + 1) (x^{2} - x \cdot 1 + 1^{2})) = 0$

Step 10

Factor.

Simplify.

Multiply $- 1$ by $1$ .

$(x - 4) (x^{2} + 4 x + 16) ((x + 1) (x^{2} - x + 1^{2})) = 0$

One to any power is one.

$(x - 4) (x^{2} + 4 x + 16) ((x + 1) (x^{2} - x + 1)) = 0$

Remove unnecessary parentheses.

$(x - 4) (x^{2} + 4 x + 16) (x + 1) (x^{2} - x + 1) = 0$

Step 11

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

$x - 4 = 0$

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

$x + 1 = 0$

$x^{2} - x + 1 = 0$

Step 12

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 13

Set $x^{2} + 4 x + 16$ equal to $0$ and solve for $x$ .

Set $x^{2} + 4 x + 16$ equal to $0$ .

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

Solve $x^{2} + 4 x + 16 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 4$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 16)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 16$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

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

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$x = - 2 \pm 2 i \sqrt{3}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 16$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

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

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$x = - 2 \pm 2 i \sqrt{3}$

Change the $\pm$ to $+$ .

$x = - 2 + 2 i \sqrt{3}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 16$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

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

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$x = - 2 \pm 2 i \sqrt{3}$

Change the $\pm$ to $-$ .

$x = - 2 - 2 i \sqrt{3}$

The final answer is the combination of both solutions.

$x = - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}$

Step 14

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 15

Set $x^{2} - x + 1$ equal to $0$ and solve for $x$ .

Set $x^{2} - x + 1$ equal to $0$ .

$x^{2} - x + 1 = 0$

Solve $x^{2} - x + 1 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 1$ .

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Subtract $4$ from $1$ .

$x = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 1$ .

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Subtract $4$ from $1$ .

$x = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{1 + i \sqrt{3}}{2}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 1$ .

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Subtract $4$ from $1$ .

$x = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{1 - i \sqrt{3}}{2}$

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

Step 16

The final solution is all the values that make $(x - 4) (x^{2} + 4 x + 16) (x + 1) (x^{2} - x + 1) = 0$ true.

$x = 4, - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

Solve by Factoring x^6-63x^3-64=0

Related Questions

Support LCMGCF.com by disabling your adblocker.