Find the Roots (Zeros) f(x)=x^4+72x^2-729

$f (x) = x^{4} + 72 x^{2} - 729$

Step 1

Set $x^{4} + 72 x^{2} - 729$ equal to $0$ .

$x^{4} + 72 x^{2} - 729 = 0$

Step 2

Solve for $x$ .

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} + 72 u - 729 = 0$

$u = x^{2}$

Factor $u^{2} + 72 u - 729$ 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 $- 729$ and whose sum is $72$ .

$- 9, 81$

Write the factored form using these integers.

$(u - 9) (u + 81) = 0$

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

$u - 9 = 0$

$u + 81 = 0$

Set $u - 9$ equal to $0$ and solve for $u$ .

Set $u - 9$ equal to $0$ .

$u - 9 = 0$

Add $9$ to both sides of the equation.

$u = 9$

Set $u + 81$ equal to $0$ and solve for $u$ .

Set $u + 81$ equal to $0$ .

$u + 81 = 0$

Subtract $81$ from both sides of the equation.

$u = - 81$

The final solution is all the values that make $(u - 9) (u + 81) = 0$ true.

$u = 9, - 81$

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 9$

${(x^{2})}^{1} = - 81$

Solve the first equation for $x$ .

$x^{2} = 9$

Solve the equation for $x$ .

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

$x = \pm \sqrt{9}$

Simplify $\pm \sqrt{9}$ .

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

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

$x = \pm 3$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$

Solve the second equation for $x$ .

${(x^{2})}^{1} = - 81$

Solve the equation for $x$ .

Remove parentheses.

$x^{2} = - 81$

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

$x = \pm \sqrt{- 81}$

Simplify $\pm \sqrt{- 81}$ .

Rewrite $- 81$ as $- 1 (81)$ .

$x = \pm \sqrt{- 1 (81)}$

Rewrite $\sqrt{- 1 (81)}$ as $\sqrt{- 1} \cdot \sqrt{81}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{81}$

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

$x = \pm i \cdot \sqrt{81}$

Rewrite $81$ as $9^{2}$ .

$x = \pm i \cdot \sqrt{9^{2}}$

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

$x = \pm i \cdot 9$

Move $9$ to the left of $i$ .

$x = \pm 9 i$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 9 i$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 9 i$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 9 i, - 9 i$

The solution to $x^{4} + 72 x^{2} - 729 = 0$ is $x = 3, - 3, 9 i, - 9 i$ .

$x = 3, - 3, 9 i, - 9 i$

Step 3

Find the Roots (Zeros) f(x)=x^4+72x^2-729

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