Algebra Examples

${| x + 7 |}^{2} - 3 | x + 7 | - 4 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = | x + 7 |$ . Substitute $u$ for all occurrences of $| x + 7 |$ .

$u^{2} - 3 u - 4 = 0$

Step 1.2

Factor $u^{2} - 3 u - 4$ using the AC method.

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Step 1.2.1

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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 1.2.2

Write the factored form using these integers.

$(u - 4) (u + 1) = 0$

Step 1.3

Replace all occurrences of $u$ with $| x + 7 |$ .

$(| x + 7 | - 4) (| x + 7 | + 1) = 0$

Step 2

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

$| x + 7 | - 4 = 0$

$| x + 7 | + 1 = 0$

Step 3

Set $| x + 7 | - 4$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $| x + 7 | - 4$ equal to $0$ .

$| x + 7 | - 4 = 0$

Step 3.2

Solve $| x + 7 | - 4 = 0$ for $x$ .

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Step 3.2.1

Add $4$ to both sides of the equation.

$| x + 7 | = 4$

Step 3.2.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x + 7 = \pm 4$

Step 3.2.3

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

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Step 3.2.3.1

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

$x + 7 = 4$

Step 3.2.3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 3.2.3.2.1

Subtract $7$ from both sides of the equation.

$x = 4 - 7$

Step 3.2.3.2.2

Subtract $7$ from $4$ .

$x = - 3$

Step 3.2.3.3

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

$x + 7 = - 4$

Step 3.2.3.4

Move all terms not containing $x$ to the right side of the equation.

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Step 3.2.3.4.1

Subtract $7$ from both sides of the equation.

$x = - 4 - 7$

Step 3.2.3.4.2

Subtract $7$ from $- 4$ .

$x = - 11$

Step 3.2.3.5

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

$x = - 3, - 11$

Step 4

Set $| x + 7 | + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $| x + 7 | + 1$ equal to $0$ .

$| x + 7 | + 1 = 0$

Step 4.2

Solve $| x + 7 | + 1 = 0$ for $x$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$| x + 7 | = - 1$

Step 4.2.2

There is no value of $x$ that makes the equation be true since an absolute value can never be negative.

No solution

Step 5

The final solution is all the values that make $(| x + 7 | - 4) (| x + 7 | + 1) = 0$ true.

$x = - 3, - 11$

Step 6