Algebra Examples

$x^{2} + 2 x - 7 = {(x + a)}^{2} + b$

Step 1

Rewrite the equation as ${(x + a)}^{2} + b = x^{2} + 2 x - 7$ .

${(x + a)}^{2} + b = x^{2} + 2 x - 7$

Step 2

Subtract $b$ from both sides of the equation.

${(x + a)}^{2} = x^{2} + 2 x - 7 - b$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + a = \pm \sqrt{x^{2} + 2 x - 7 - b}$

Step 4

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

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Step 4.1

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

$x + a = \sqrt{x^{2} + 2 x - 7 - b}$

Step 4.2

Subtract $x$ from both sides of the equation.

$a = \sqrt{x^{2} + 2 x - 7 - b} - x$

Step 4.3

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

$x + a = - \sqrt{x^{2} + 2 x - 7 - b}$

Step 4.4

Subtract $x$ from both sides of the equation.

$a = - \sqrt{x^{2} + 2 x - 7 - b} - x$

Step 4.5

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

$a = \sqrt{x^{2} + 2 x - 7 - b} - x$

$a = - \sqrt{x^{2} + 2 x - 7 - b} - x$

$a = \sqrt{x^{2} + 2 x - 7 - b} - x$

$a = - \sqrt{x^{2} + 2 x - 7 - b} - x$