Find the Roots (Zeros) f(x)=x^4-24x^2-25

$f (x) = x^{4} - 24 x^{2} - 25$

Step 1

Set $x^{4} - 24 x^{2} - 25$ equal to $0$ .

$x^{4} - 24 x^{2} - 25 = 0$

Step 2

Solve for $x$ .

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

$u^{2} - 24 u - 25 = 0$

$u = x^{2}$

Factor $u^{2} - 24 u - 25$ 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 $- 25$ and whose sum is $- 24$ .

$- 25, 1$

Write the factored form using these integers.

$(u - 25) (u + 1) = 0$

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

$u - 25 = 0$

$u + 1 = 0$

Set $u - 25$ equal to $0$ and solve for $u$ .

Set $u - 25$ equal to $0$ .

$u - 25 = 0$

Add $25$ to both sides of the equation.

$u = 25$

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

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

$u + 1 = 0$

Subtract $1$ from both sides of the equation.

$u = - 1$

The final solution is all the values that make $(u - 25) (u + 1) = 0$ true.

$u = 25, - 1$

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

$x^{2} = 25$

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

Solve the first equation for $x$ .

$x^{2} = 25$

Solve the equation for $x$ .

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

$x = \pm \sqrt{25}$

Simplify $\pm \sqrt{25}$ .

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2}}$

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

$x = \pm 5$

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.

$x = 5$

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

$x = - 5$

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

$x = 5, - 5$

Solve the second equation for $x$ .

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

Solve the equation for $x$ .

Remove parentheses.

$x^{2} = - 1$

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

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

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

$x = \pm i$

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.

$x = i$

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

$x = - i$

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

$x = i, - i$

The solution to $x^{4} - 24 x^{2} - 25 = 0$ is $x = 5, - 5, i, - i$ .

$x = 5, - 5, i, - i$

Step 3

Find the Roots (Zeros) f(x)=x^4-24x^2-25

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