Algebra Examples

$\frac{x + 2}{X + 4} = \frac{x - 2}{x + 2}$

Step 1

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.

$(x + 2) (x + 2) = (X + 4) (x - 2)$

Step 2

Solve the equation for $x$ .

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Step 2.1

Simplify $(x + 2) (x + 2)$ .

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Step 2.1.1

Rewrite.

$0 + 0 + (x + 2) (x + 2) = (X + 4) (x - 2)$

Step 2.1.2

Simplify by adding zeros.

$(x + 2) (x + 2) = (X + 4) (x - 2)$

Step 2.1.3

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 2.1.3.1

Apply the distributive property.

$x (x + 2) + 2 (x + 2) = (X + 4) (x - 2)$

Step 2.1.3.2

Apply the distributive property.

$x \cdot x + x \cdot 2 + 2 (x + 2) = (X + 4) (x - 2)$

Step 2.1.3.3

Apply the distributive property.

$x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2 = (X + 4) (x - 2)$

Step 2.1.4

Simplify and combine like terms.

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Step 2.1.4.1

Simplify each term.

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Step 2.1.4.1.1

Multiply $x$ by $x$ .

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

Step 2.1.4.1.2

Move $2$ to the left of $x$ .

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

Step 2.1.4.1.3

Multiply $2$ by $2$ .

$x^{2} + 2 x + 2 x + 4 = (X + 4) (x - 2)$

Step 2.1.4.2

Add $2 x$ and $2 x$ .

$x^{2} + 4 x + 4 = (X + 4) (x - 2)$

Step 2.2

Simplify $(X + 4) (x - 2)$ .

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Step 2.2.1

Expand $(X + 4) (x - 2)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

$x^{2} + 4 x + 4 = X (x - 2) + 4 (x - 2)$

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

$x^{2} + 4 x + 4 = X x + X \cdot - 2 + 4 x + 4 \cdot - 2$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Move $- 2$ to the left of $X$ .

$x^{2} + 4 x + 4 = X x - 2 \cdot X + 4 x + 4 \cdot - 2$

Step 2.2.2.2

Multiply $4$ by $- 2$ .

$x^{2} + 4 x + 4 = X x - 2 X + 4 x - 8$

Step 2.3

Move all terms containing $x$ to the left side of the equation.

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Step 2.3.1

Subtract $X x$ from both sides of the equation.

$x^{2} + 4 x + 4 - X x = - 2 X + 4 x - 8$

Step 2.3.2

Subtract $4 x$ from both sides of the equation.

$x^{2} + 4 x + 4 - X x - 4 x = - 2 X - 8$

Step 2.3.3

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

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Step 2.3.3.1

Subtract $4 x$ from $4 x$ .

$x^{2} + 4 - X x + 0 = - 2 X - 8$

Step 2.3.3.2

Add $x^{2} + 4 - X x$ and $0$ .

$x^{2} + 4 - X x = - 2 X - 8$

Step 2.4

Move all terms to the left side of the equation and simplify.

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Step 2.4.1

Move all the expressions to the left side of the equation.

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Step 2.4.1.1

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

$x^{2} + 4 - X x + 2 X = - 8$

Step 2.4.1.2

Add $8$ to both sides of the equation.

$x^{2} + 4 - X x + 2 X + 8 = 0$

Step 2.4.2

Add $4$ and $8$ .

$x^{2} - X x + 2 X + 12 = 0$

Step 2.5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.6

Substitute the values $a = 1$ , $b = - X$ , and $c = 2 X + 12$ into the quadratic formula and solve for $x$ .

$\frac{X \pm \sqrt{{(- X)}^{2} - 4 \cdot (1 \cdot (2 X + 12))}}{2 \cdot 1}$

Step 2.7

Simplify.

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Apply the product rule to $- X$ .

$x = \frac{X \pm \sqrt{{(- 1)}^{2} X^{2} - 4 \cdot 1 \cdot (2 X + 12)}}{2 \cdot 1}$

Step 2.7.1.2

Raise $- 1$ to the power of $2$ .

$x = \frac{X \pm \sqrt{1 X^{2} - 4 \cdot 1 \cdot (2 X + 12)}}{2 \cdot 1}$

Step 2.7.1.3

Multiply $X^{2}$ by $1$ .

$x = \frac{X \pm \sqrt{X^{2} - 4 \cdot 1 \cdot (2 X + 12)}}{2 \cdot 1}$

Step 2.7.1.4

Multiply $- 4$ by $1$ .

$x = \frac{X \pm \sqrt{X^{2} - 4 \cdot (2 X + 12)}}{2 \cdot 1}$

Step 2.7.1.5

Apply the distributive property.

$x = \frac{X \pm \sqrt{X^{2} - 4 (2 X) - 4 \cdot 12}}{2 \cdot 1}$

Step 2.7.1.6

Multiply $2$ by $- 4$ .

$x = \frac{X \pm \sqrt{X^{2} - 8 X - 4 \cdot 12}}{2 \cdot 1}$

Step 2.7.1.7

Multiply $- 4$ by $12$ .

$x = \frac{X \pm \sqrt{X^{2} - 8 X - 48}}{2 \cdot 1}$

Step 2.7.1.8

Factor $X^{2} - 8 X - 48$ using the AC method.

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Step 2.7.1.8.1

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 $- 48$ and whose sum is $- 8$ .

$- 12, 4$

Step 2.7.1.8.2

Write the factored form using these integers.

$x = \frac{X \pm \sqrt{(X - 12) (X + 4)}}{2 \cdot 1}$

Step 2.7.2

Multiply $2$ by $1$ .

$x = \frac{X \pm \sqrt{(X - 12) (X + 4)}}{2}$

Step 2.8

The final answer is the combination of both solutions.

$x = \frac{X + \sqrt{(X - 12) (X + 4)}}{2}$

$x = \frac{X - \sqrt{(X - 12) (X + 4)}}{2}$

$x = \frac{X + \sqrt{(X - 12) (X + 4)}}{2}$

$x = \frac{X - \sqrt{(X - 12) (X + 4)}}{2}$