Find All Complex Solutions |x^2+x-28|=28

$| x^{2} + x - 28 | = 28$

Step 1

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

$x^{2} + x - 28 = \pm 28$

Step 2

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

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

$x^{2} + x - 28 = 28$

Subtract $28$ from both sides of the equation.

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

Subtract $28$ from $- 28$ .

$x^{2} + x - 56 = 0$

Factor $x^{2} + x - 56$ using the AC method.

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 $- 56$ and whose sum is $1$ .

$- 7, 8$

Write the factored form using these integers.

$(x - 7) (x + 8) = 0$

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

$x - 7 = 0$

$x + 8 = 0$

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

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

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Set $x + 8$ equal to $0$ and solve for $x$ .

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

The final solution is all the values that make $(x - 7) (x + 8) = 0$ true.

$x = 7, - 8$

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

$x^{2} + x - 28 = - 28$

Add $28$ to both sides of the equation.

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

Combine the opposite terms in $x^{2} + x - 28 + 28$ .

Add $- 28$ and $28$ .

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

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

$x^{2} + x = 0$

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

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

$x \cdot x + x = 0$

Raise $x$ to the power of $1$ .

$x \cdot x + x = 0$

Factor $x$ out of $x^{1}$ .

$x \cdot x + x \cdot 1 = 0$

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$x (x + 1) = 0$

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

$x = 0$

$x + 1 = 0$

Set $x$ equal to $0$ .

$x = 0$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

$x = 0, - 1$

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

$x = 7, - 8, 0, - 1$

Find All Complex Solutions |x^2+x-28|=28

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