Algebra Examples

$y = {(x + 4)}^{2} - 1$ and $y = - 3 x - 13$

Step 1

Eliminate the equal sides of each equation and combine.

${(x + 4)}^{2} - 1 = - 3 x - 13$

Step 2

Solve ${(x + 4)}^{2} - 1 = - 3 x - 13$ for $x$ .

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Step 2.1

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

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Step 2.1.1

Add $3 x$ to both sides of the equation.

${(x + 4)}^{2} - 1 + 3 x = - 13$

Step 2.1.2

Simplify each term.

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Step 2.1.2.1

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

$(x + 4) (x + 4) - 1 + 3 x = - 13$

Step 2.1.2.2

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

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Step 2.1.2.2.1

Apply the distributive property.

$x (x + 4) + 4 (x + 4) - 1 + 3 x = - 13$

Step 2.1.2.2.2

Apply the distributive property.

$x \cdot x + x \cdot 4 + 4 (x + 4) - 1 + 3 x = - 13$

Step 2.1.2.2.3

Apply the distributive property.

$x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4 - 1 + 3 x = - 13$

Step 2.1.2.3

Simplify and combine like terms.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 4 + 4 x + 4 \cdot 4 - 1 + 3 x = - 13$

Step 2.1.2.3.1.2

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

$x^{2} + 4 \cdot x + 4 x + 4 \cdot 4 - 1 + 3 x = - 13$

Step 2.1.2.3.1.3

Multiply $4$ by $4$ .

$x^{2} + 4 x + 4 x + 16 - 1 + 3 x = - 13$

Step 2.1.2.3.2

Add $4 x$ and $4 x$ .

$x^{2} + 8 x + 16 - 1 + 3 x = - 13$

Step 2.1.3

Add $8 x$ and $3 x$ .

$x^{2} + 11 x + 16 - 1 = - 13$

Step 2.1.4

Subtract $1$ from $16$ .

$x^{2} + 11 x + 15 = - 13$

Step 2.2

Add $13$ to both sides of the equation.

$x^{2} + 11 x + 15 + 13 = 0$

Step 2.3

Add $15$ and $13$ .

$x^{2} + 11 x + 28 = 0$

Step 2.4

Factor $x^{2} + 11 x + 28$ using the AC method.

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Step 2.4.1

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 $28$ and whose sum is $11$ .

$4, 7 + y = - 3 x - 13$

Step 2.4.2

Write the factored form using these integers.

$(x + 4) (x + 7) = 0$

Step 2.5

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

$x + 4 = 0$

$x + 7 = 0 + y = - 3 x - 13$

Step 2.6

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

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Step 2.6.1

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

$x + 4 = 0$

Step 2.6.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.7

Set $x + 7$ equal to $0$ and solve for $x$ .

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Step 2.7.1

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 2.7.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 2.8

The final solution is all the values that make $(x + 4) (x + 7) = 0$ true.

$x = - 4, - 7$

Step 3

Evaluate $y$ when $x = - 4$ .

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Step 3.1

Substitute $- 4$ for $x$ .

$y = - 3 \cdot - 4 - 13$

Step 3.2

Substitute $- 4$ for $x$ in $y = - 3 \cdot - 4 - 13$ and solve for $y$ .

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Step 3.2.1

Remove parentheses.

$y = - 3 \cdot - 4 - 13$

Step 3.2.2

Simplify $- 3 \cdot - 4 - 13$ .

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Step 3.2.2.1

Multiply $- 3$ by $- 4$ .

$y = 12 - 13$

Step 3.2.2.2

Subtract $13$ from $12$ .

$y = - 1$

Step 4

Evaluate $y$ when $x = - 7$ .

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Step 4.1

Substitute $- 7$ for $x$ .

$y = - 3 \cdot - 7 - 13$

Step 4.2

Substitute $- 7$ for $x$ in $y = - 3 \cdot - 7 - 13$ and solve for $y$ .

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Step 4.2.1

Remove parentheses.

$y = - 3 \cdot - 7 - 13$

Step 4.2.2

Simplify $- 3 \cdot - 7 - 13$ .

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Step 4.2.2.1

Multiply $- 3$ by $- 7$ .

$y = 21 - 13$

Step 4.2.2.2

Subtract $13$ from $21$ .

$y = 8$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- 4, - 1)$

$(- 7, 8)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 4, - 1), (- 7, 8)$

Equation Form:

$x = - 4, y = - 1$

$x = - 7, y = 8$

Step 7