Algebra Examples

$x^{4} - 6 x^{2} + 5 = 0$

Step 1

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

$u^{2} - 6 u + 5 = 0$

$u = x^{2}$

Step 2

Factor $u^{2} - 6 u + 5$ using the AC method.

Tap for more steps...

Step 2.1

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 $5$ and whose sum is $- 6$ .

$- 5, - 1$

Step 2.2

Write the factored form using these integers.

$(u - 5) (u - 1) = 0$

Step 3

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

$u - 5 = 0$

$u - 1 = 0$

Step 4

Set $u - 5$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 4.1

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

$u - 5 = 0$

Step 4.2

Add $5$ to both sides of the equation.

$u = 5$

Step 5

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

Tap for more steps...

Step 5.1

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

$u - 1 = 0$

Step 5.2

Add $1$ to both sides of the equation.

$u = 1$

Step 6

The final solution is all the values that make $(u - 5) (u - 1) = 0$ true.

$u = 5, 1$

Step 7

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

$x^{2} = 5$

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

Step 8

Solve the first equation for $x$ .

$x^{2} = 5$

Step 9

Solve the equation for $x$ .

Tap for more steps...

Step 9.1

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

$x = \pm \sqrt{5}$

Step 9.2

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

Tap for more steps...

Step 9.2.1

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

$x = \sqrt{5}$

Step 9.2.2

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

$x = - \sqrt{5}$

Step 9.2.3

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

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

Step 10

Solve the second equation for $x$ .

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

Step 11

Solve the equation for $x$ .

Tap for more steps...

Step 11.1

Remove parentheses.

$x^{2} = 1$

Step 11.2

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

$x = \pm \sqrt{1}$

Step 11.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 11.4

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

Tap for more steps...

Step 11.4.1

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

$x = 1$

Step 11.4.2

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

$x = - 1$

Step 11.4.3

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

$x = 1, - 1$

Step 12

The solution to $x^{4} - 6 x^{2} + 5 = 0$ is $x = \sqrt{5}, - \sqrt{5}, 1, - 1$ .

$x = \sqrt{5}, - \sqrt{5}, 1, - 1$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{5}, - \sqrt{5}, 1, - 1$

Decimal Form:

$x = 2.23606797 \dots, - 2.23606797 \dots, 1, - 1$