Algebra Examples

x^{4} - 4 x^{2} + 4 = 0

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

Step 1

Substitute

u = x^{2}

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

u^{2} - 4 u + 4 = 0

$u^{2} - 4 u + 4 = 0$

u = x^{2}

$u = x^{2}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

u^{2} - 4 u + 2^{2} = 0

$u^{2} - 4 u + 2^{2} = 0$

Step 2.2

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

4 u = 2 \cdot u \cdot 2

$4 u = 2 \cdot u \cdot 2$

Step 2.3

Rewrite the polynomial.

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

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

Step 2.4

Factor using the perfect square trinomial rule

a^{2} - 2 a b + b^{2} = {(a - b)}^{2}

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

a = u

$a = u$ and

b = 2

$b = 2$ .

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

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

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

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

Step 3

Set the

u - 2

$u - 2$ equal to

0

$0$ .

u - 2 = 0

$u - 2 = 0$

Step 4

Add

2

$2$ to both sides of the equation.

u = 2

$u = 2$

Step 5

Substitute the real value of

u = x^{2}

$u = x^{2}$ back into the solved equation.

x^{2} = 2

$x^{2} = 2$

Step 6

Solve the equation for

x

$x$ .

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Step 6.1

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

x = \pm \sqrt{2}

$x = \pm \sqrt{2}$

Step 6.2

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

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Step 6.2.1

First, use the positive value of the

\pm

$\pm$ to find the first solution.

x = \sqrt{2}

$x = \sqrt{2}$

Step 6.2.2

Next, use the negative value of the

\pm

$\pm$ to find the second solution.

x = - \sqrt{2}

$x = - \sqrt{2}$

Step 6.2.3

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

x = \sqrt{2}, - \sqrt{2}

$x = \sqrt{2}, - \sqrt{2}$

x = \sqrt{2}, - \sqrt{2}

$x = \sqrt{2}, - \sqrt{2}$

x = \sqrt{2}, - \sqrt{2}

$x = \sqrt{2}, - \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

x = \sqrt{2}, - \sqrt{2}

$x = \sqrt{2}, - \sqrt{2}$

Decimal Form:

x = 1.41421356 \dots, - 1.41421356 \dots

$x = 1.41421356 \dots, - 1.41421356 \dots$

Step 8