Pre-Algebra Examples

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

Step 1

Subtract $x^{4}$ from both sides of the equation.

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

Step 2

Subtract $25$ from both sides of the equation.

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

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $- 1$ out of $10 x^{2} - x^{4} - 25$ .

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Step 3.1.1

Reorder $10 x^{2}$ and $- x^{4}$ .

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

Step 3.1.2

Factor $- 1$ out of $- x^{4}$ .

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

Step 3.1.3

Factor $- 1$ out of $10 x^{2}$ .

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

Step 3.1.4

Rewrite $- 25$ as $- 1 (25)$ .

$- (x^{4}) - (- 10 x^{2}) - 1 \cdot 25 = 0$

Step 3.1.5

Factor $- 1$ out of $- (x^{4}) - (- 10 x^{2})$ .

$- (x^{4} - 10 x^{2}) - 1 \cdot 25 = 0$

Step 3.1.6

Factor $- 1$ out of $- (x^{4} - 10 x^{2}) - 1 (25)$ .

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

Step 3.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 3.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

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

Step 3.4

Factor using the perfect square rule.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Rewrite the polynomial.

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

Step 3.4.4

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 3.5

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 4

Divide each term in $- {(x^{2} - 5)}^{2} = 0$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- {(x^{2} - 5)}^{2} = 0$ by $- 1$ .

$\frac{- {(x^{2} - 5)}^{2}}{- 1} = \frac{0}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{{(x^{2} - 5)}^{2}}{1} = \frac{0}{- 1}$

Step 4.2.2

Divide ${(x^{2} - 5)}^{2}$ by $1$ .

${(x^{2} - 5)}^{2} = \frac{0}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 1$ .

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

Step 5

Set the $x^{2} - 5$ equal to $0$ .

$x^{2} - 5 = 0$

Step 6

Solve for $x$ .

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Step 6.1

Add $5$ to both sides of the equation.

$x^{2} = 5$

Step 6.2

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

$x = \pm \sqrt{5}$

Step 6.3

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

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Step 6.3.1

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

$x = \sqrt{5}$

Step 6.3.2

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

$x = - \sqrt{5}$

Step 6.3.3

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:

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