Algebra Examples

$X^{2} - 2 x + 5 = 0$

Step 1

Subtract $5$ from both sides of the equation.

$X^{2} - 2 x = - 5$

Step 2

Rewrite $X^{2} - 2 x$ as a difference of squares.

$X^{2} - {\sqrt{2 x}}^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = X$ and $b = \sqrt{2 x}$ .

$(X + \sqrt{2 x}) (X - \sqrt{2 x})$

Step 4

Simplify $(X + \sqrt{2 x}) (X - \sqrt{2 x})$ .

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Step 4.1

Expand $(X + \sqrt{2 x}) (X - \sqrt{2 x})$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$X (X - \sqrt{2 x}) + \sqrt{2 x} (X - \sqrt{2 x}) = - 5$

Step 4.1.2

Apply the distributive property.

$X \cdot X + X (- \sqrt{2 x}) + \sqrt{2 x} (X - \sqrt{2 x}) = - 5$

Step 4.1.3

Apply the distributive property.

$X \cdot X + X (- \sqrt{2 x}) + \sqrt{2 x} X + \sqrt{2 x} (- \sqrt{2 x}) = - 5$

Step 4.2

Simplify terms.

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Step 4.2.1

Combine the opposite terms in $X \cdot X + X (- \sqrt{2 x}) + \sqrt{2 x} X + \sqrt{2 x} (- \sqrt{2 x})$ .

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Step 4.2.1.1

Reorder the factors in the terms $X (- \sqrt{2 x})$ and $\sqrt{2 x} X$ .

$X \cdot X - X \sqrt{2 x} + X \sqrt{2 x} + \sqrt{2 x} (- \sqrt{2 x}) = - 5$

Step 4.2.1.2

Add $- X \sqrt{2 x}$ and $X \sqrt{2 x}$ .

$X \cdot X + 0 + \sqrt{2 x} (- \sqrt{2 x}) = - 5$

Step 4.2.1.3

Add $X \cdot X$ and $0$ .

$X \cdot X + \sqrt{2 x} (- \sqrt{2 x}) = - 5$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Multiply $X$ by $X$ .

$X^{2} + \sqrt{2 x} (- \sqrt{2 x}) = - 5$

Step 4.2.2.2

Rewrite using the commutative property of multiplication.

$X^{2} - \sqrt{2 x} \sqrt{2 x} = - 5$

Step 4.2.2.3

Multiply $- \sqrt{2 x} \sqrt{2 x}$ .

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Step 4.2.2.3.1

Raise $\sqrt{2 x}$ to the power of $1$ .

$X^{2} - ({\sqrt{2 x}}^{1} \sqrt{2 x}) = - 5$

Step 4.2.2.3.2

Raise $\sqrt{2 x}$ to the power of $1$ .

$X^{2} - ({\sqrt{2 x}}^{1} {\sqrt{2 x}}^{1}) = - 5$

Step 4.2.2.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$X^{2} - {\sqrt{2 x}}^{1 + 1} = - 5$

Step 4.2.2.3.4

Add $1$ and $1$ .

$X^{2} - {\sqrt{2 x}}^{2} = - 5$

Step 4.2.2.4

Rewrite ${\sqrt{2 x}}^{2}$ as $2 x$ .

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Step 4.2.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x}$ as ${(2 x)}^{\frac{1}{2}}$ .

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

Step 4.2.2.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.2.2.4.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 4.2.2.4.4

Cancel the common factor of $2$ .

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Step 4.2.2.4.4.1

Cancel the common factor.

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

Step 4.2.2.4.4.2

Rewrite the expression.

$X^{2} - {(2 x)}^{1} = - 5$

Step 4.2.2.4.5

Simplify.

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

Step 4.2.2.5

Multiply $2$ by $- 1$ .

$X^{2} - 2 x = - 5$

Step 5

Add $2 x$ to both sides of the equation.

$X^{2} = - 5 + 2 x$

Step 6

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

$X = \pm \sqrt{- 5 + 2 x}$

Step 7

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

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Step 7.1

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

$X = \sqrt{- 5 + 2 x}$

Step 7.2

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

$X = - \sqrt{- 5 + 2 x}$

Step 7.3

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

$X = \sqrt{- 5 + 2 x}$

$X = - \sqrt{- 5 + 2 x}$

$X = \sqrt{- 5 + 2 x}$

$X = - \sqrt{- 5 + 2 x}$