Solve for x x^4-10x^2+25=0

$x^{4} - 10 x^{2} + 25 = 0$

Step 1

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

$u^{2} - 10 u + 25 = 0$

$u = x^{2}$

Step 2

Factor using the perfect square rule.

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

$u^{2} - 10 u + 5^{2} = 0$

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

$10 u = 2 \cdot u \cdot 5$

Rewrite the polynomial.

$u^{2} - 2 \cdot u \cdot 5 + 5^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 5$ .

${(u - 5)}^{2} = 0$

Step 3

Set the $u - 5$ equal to $0$ .

$u - 5 = 0$

Step 4

Add $5$ to both sides of the equation.

$u = 5$

Step 5

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

$x^{2} = 5$

Step 6

Solve the equation for $x$ .

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

$x = \pm \sqrt{5}$

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.

$x = \sqrt{5}$

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

$x = - \sqrt{5}$

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

$x = \sqrt{5}, - \sqrt{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{5}, - \sqrt{5}$

Decimal Form:

$x = 2.23606797 \dots, - 2.23606797 \dots$