Precalculus Examples

$\frac{1}{2} = \frac{x^{2} + 2}{x + 4}$

Step 1

Rewrite the equation as $\frac{x^{2} + 2}{x + 4} = \frac{1}{2}$ .

$\frac{x^{2} + 2}{x + 4} = \frac{1}{2}$

Step 2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$(x^{2} + 2) \cdot 2 = (x + 4) \cdot 1$

Step 3

Solve the equation for $x$ .

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Step 3.1

Simplify $(x^{2} + 2) \cdot 2$ .

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Step 3.1.1

Rewrite.

$0 + 0 + (x^{2} + 2) \cdot 2 = (x + 4) \cdot 1$

Step 3.1.2

Simplify by adding zeros.

$(x^{2} + 2) \cdot 2 = (x + 4) \cdot 1$

Step 3.1.3

Apply the distributive property.

$x^{2} \cdot 2 + 2 \cdot 2 = (x + 4) \cdot 1$

Step 3.1.4

Simplify the expression.

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Step 3.1.4.1

Move $2$ to the left of $x^{2}$ .

$2 \cdot x^{2} + 2 \cdot 2 = (x + 4) \cdot 1$

Step 3.1.4.2

Multiply $2$ by $2$ .

$2 x^{2} + 4 = (x + 4) \cdot 1$

Step 3.2

Multiply $x + 4$ by $1$ .

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

Step 3.3

Subtract $x$ from both sides of the equation.

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

Step 3.4

Subtract $4$ from both sides of the equation.

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

Step 3.5

Combine the opposite terms in $2 x^{2} + 4 - x - 4$ .

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Step 3.5.1

Subtract $4$ from $4$ .

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

Step 3.5.2

Add $2 x^{2} - x$ and $0$ .

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

Step 3.6

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

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Step 3.6.1

Factor $x$ out of $2 x^{2}$ .

$x (2 x) - x = 0$

Step 3.6.2

Factor $x$ out of $- x$ .

$x (2 x) + x \cdot - 1 = 0$

Step 3.6.3

Factor $x$ out of $x (2 x) + x \cdot - 1$ .

$x (2 x - 1) = 0$

Step 3.7

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

$x = 0$

$2 x - 1 = 0$

Step 3.8

Set $x$ equal to $0$ .

$x = 0$

Step 3.9

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 3.9.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 3.9.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 3.9.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 3.9.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 3.9.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 3.9.2.2.2

Simplify the left side.

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Step 3.9.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.9.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 3.9.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 3.10

The final solution is all the values that make $x (2 x - 1) = 0$ true.

$x = 0, \frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{1}{2}$

Decimal Form:

$x = 0, 0.5$