Algebra Examples

$y - x = 4$ $y = \frac{x^{2} + 2 x - 8}{4}$

Step 1

Add $x$ to both sides of the equation.

$y = 4 + x$

$y = \frac{x^{2} + 2 x - 8}{4}$

Step 2

Replace all occurrences of $y$ with $4 + x$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $y = 4 + x$ with $4 + x$ .

$4 + x = 4 + x$

$y = \frac{x^{2} + 2 x - 8}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Remove parentheses.

$4 + x = 4 + x$

$y = \frac{x^{2} + 2 x - 8}{4}$

$4 + x = 4 + x$

$y = \frac{x^{2} + 2 x - 8}{4}$

Step 2.3

Replace all occurrences of $y$ in $y = \frac{x^{2} + 2 x - 8}{4}$ with $4 + x$ .

$4 + x = \frac{x^{2} + 2 x - 8}{4}$

$4 + x = 4 + x$

Step 2.4

Simplify $4 + x = \frac{x^{2} + 2 x - 8}{4}$ .

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Remove parentheses.

$4 + x = \frac{x^{2} + 2 x - 8}{4}$

$4 + x = 4 + x$

$4 + x = \frac{x^{2} + 2 x - 8}{4}$

$4 + x = 4 + x$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Factor $x^{2} + 2 x - 8$ using the AC method.

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Step 2.4.2.1.1

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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 2.4.2.1.2

Write the factored form using these integers.

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

$4 + x = 4 + x$

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

$4 + x = 4 + x$

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

$4 + x = 4 + x$

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

$4 + x = 4 + x$

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

$4 + x = 4 + x$

Step 3

Remove any equations from the system that are always true.

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

Step 4

Solve for $x$ in $4 + x = \frac{(x - 2) (x + 4)}{4}$ .

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Step 4.1

Multiply both sides by $4$ .

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

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $(4 + x) \cdot 4$ .

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Step 4.2.1.1.1

Apply the distributive property.

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

Step 4.2.1.1.2

Simplify the expression.

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Step 4.2.1.1.2.1

Multiply $4$ by $4$ .

$16 + x \cdot 4 = \frac{(x - 2) (x + 4)}{4} \cdot 4$

Step 4.2.1.1.2.2

Move $4$ to the left of $x$ .

$16 + 4 \cdot x = \frac{(x - 2) (x + 4)}{4} \cdot 4$

Step 4.2.1.1.2.3

Reorder $16$ and $4 x$ .

$4 x + 16 = \frac{(x - 2) (x + 4)}{4} \cdot 4$

$4 x + 16 = \frac{(x - 2) (x + 4)}{4} \cdot 4$

$4 x + 16 = \frac{(x - 2) (x + 4)}{4} \cdot 4$

$4 x + 16 = \frac{(x - 2) (x + 4)}{4} \cdot 4$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $\frac{(x - 2) (x + 4)}{4} \cdot 4$ .

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Step 4.2.2.1.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1.1

Cancel the common factor.

$4 x + 16 = \frac{(x - 2) (x + 4)}{4} \cdot 4$

Step 4.2.2.1.1.2

Rewrite the expression.

$4 x + 16 = (x - 2) (x + 4)$

$4 x + 16 = (x - 2) (x + 4)$

Step 4.2.2.1.2

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

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Step 4.2.2.1.2.1

Apply the distributive property.

$4 x + 16 = x (x + 4) - 2 (x + 4)$

Step 4.2.2.1.2.2

Apply the distributive property.

$4 x + 16 = x \cdot x + x \cdot 4 - 2 (x + 4)$

Step 4.2.2.1.2.3

Apply the distributive property.

$4 x + 16 = x \cdot x + x \cdot 4 - 2 x - 2 \cdot 4$

$4 x + 16 = x \cdot x + x \cdot 4 - 2 x - 2 \cdot 4$

Step 4.2.2.1.3

Simplify and combine like terms.

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Step 4.2.2.1.3.1

Simplify each term.

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Step 4.2.2.1.3.1.1

Multiply $x$ by $x$ .

$4 x + 16 = x^{2} + x \cdot 4 - 2 x - 2 \cdot 4$

Step 4.2.2.1.3.1.2

Move $4$ to the left of $x$ .

$4 x + 16 = x^{2} + 4 \cdot x - 2 x - 2 \cdot 4$

Step 4.2.2.1.3.1.3

Multiply $- 2$ by $4$ .

$4 x + 16 = x^{2} + 4 x - 2 x - 8$

$4 x + 16 = x^{2} + 4 x - 2 x - 8$

Step 4.2.2.1.3.2

Subtract $2 x$ from $4 x$ .

$4 x + 16 = x^{2} + 2 x - 8$

$4 x + 16 = x^{2} + 2 x - 8$

$4 x + 16 = x^{2} + 2 x - 8$

$4 x + 16 = x^{2} + 2 x - 8$

$4 x + 16 = x^{2} + 2 x - 8$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 2 x - 8 = 4 x + 16$

Step 4.3.2

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

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Step 4.3.2.1

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

$x^{2} + 2 x - 8 - 4 x = 16$

Step 4.3.2.2

Subtract $4 x$ from $2 x$ .

$x^{2} - 2 x - 8 = 16$

$x^{2} - 2 x - 8 = 16$

Step 4.3.3

Subtract $16$ from both sides of the equation.

$x^{2} - 2 x - 8 - 16 = 0$

Step 4.3.4

Subtract $16$ from $- 8$ .

$x^{2} - 2 x - 24 = 0$

Step 4.3.5

Factor $x^{2} - 2 x - 24$ using the AC method.

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Step 4.3.5.1

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 $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Step 4.3.5.2

Write the factored form using these integers.

$(x - 6) (x + 4) = 0$

$(x - 6) (x + 4) = 0$

Step 4.3.6

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

$x - 6 = 0$

$x + 4 = 0$

Step 4.3.7

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 4.3.7.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 4.3.7.2

Add $6$ to both sides of the equation.

$x = 6$

$x = 6$

Step 4.3.8

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 4.3.8.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 4.3.8.2

Subtract $4$ from both sides of the equation.

$x = - 4$

$x = - 4$

Step 4.3.9

The final solution is all the values that make $(x - 6) (x + 4) = 0$ true.

$x = 6, - 4$

$x = 6, - 4$

$x = 6, - 4$

Step 5

Solve the system of equations.

$x = 6,$

Step 6

Solve the system of equations.

$x = - 4,$

Step 7

List all of the solutions.

$x = 6$

$x = - 4$

Step 8

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$