Solve (x_{0}+1)^2-4^2=(frac{1*0-x_{0}-11}{sqrt{2}})^2

$$( x _ { 0 } + 1 ) ^ { 2 } - 4 ^ { 2 } = ( \frac { 1 \times 0 - x _ { 0 } - 11 } { \sqrt { 2 } } ) ^ { 2 }$$

Solve for x_0

$$x_{0}=2\sqrt{58}+9\approx 24.231546212$$

$$x_{0}=9-2\sqrt{58}\approx -6.231546212$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x_{0}+1\right)^{2}.

$$x_{0}^{2}+2x_{0}+1-4^{2}=\left(\frac{1\times 0-x_{0}-11}{\sqrt{2}}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$x_{0}^{2}+2x_{0}+1-16=\left(\frac{1\times 0-x_{0}-11}{\sqrt{2}}\right)^{2}$$

Subtract 16 from 1 to get -15.

$$x_{0}^{2}+2x_{0}-15=\left(\frac{1\times 0-x_{0}-11}{\sqrt{2}}\right)^{2}$$

Multiply 1 and 0 to get 0.

$$x_{0}^{2}+2x_{0}-15=\left(\frac{0-x_{0}-11}{\sqrt{2}}\right)^{2}$$

Subtract 11 from 0 to get -11.

$$x_{0}^{2}+2x_{0}-15=\left(\frac{-11-x_{0}}{\sqrt{2}}\right)^{2}$$

Rationalize the denominator of \frac{-11-x_{0}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$x_{0}^{2}+2x_{0}-15=\left(\frac{\left(-11-x_{0}\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\right)^{2}$$

The square of \sqrt{2} is 2.

$$x_{0}^{2}+2x_{0}-15=\left(\frac{\left(-11-x_{0}\right)\sqrt{2}}{2}\right)^{2}$$

To raise \frac{\left(-11-x_{0}\right)\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$x_{0}^{2}+2x_{0}-15=\frac{\left(\left(-11-x_{0}\right)\sqrt{2}\right)^{2}}{2^{2}}$$

Expand \left(\left(-11-x_{0}\right)\sqrt{2}\right)^{2}.

$$x_{0}^{2}+2x_{0}-15=\frac{\left(-11-x_{0}\right)^{2}\left(\sqrt{2}\right)^{2}}{2^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-11-x_{0}\right)^{2}.

$$x_{0}^{2}+2x_{0}-15=\frac{\left(121+22x_{0}+x_{0}^{2}\right)\left(\sqrt{2}\right)^{2}}{2^{2}}$$

The square of \sqrt{2} is 2.

$$x_{0}^{2}+2x_{0}-15=\frac{\left(121+22x_{0}+x_{0}^{2}\right)\times 2}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$x_{0}^{2}+2x_{0}-15=\frac{\left(121+22x_{0}+x_{0}^{2}\right)\times 2}{4}$$

Divide \left(121+22x_{0}+x_{0}^{2}\right)\times 2 by 4 to get \left(121+22x_{0}+x_{0}^{2}\right)\times \frac{1}{2}.

$$x_{0}^{2}+2x_{0}-15=\left(121+22x_{0}+x_{0}^{2}\right)\times \frac{1}{2}$$

Use the distributive property to multiply 121+22x_{0}+x_{0}^{2} by \frac{1}{2}.

$$x_{0}^{2}+2x_{0}-15=\frac{121}{2}+11x_{0}+\frac{1}{2}x_{0}^{2}$$

Subtract \frac{121}{2} from both sides.

$$x_{0}^{2}+2x_{0}-15-\frac{121}{2}=11x_{0}+\frac{1}{2}x_{0}^{2}$$

Subtract \frac{121}{2} from -15 to get -\frac{151}{2}.

$$x_{0}^{2}+2x_{0}-\frac{151}{2}=11x_{0}+\frac{1}{2}x_{0}^{2}$$

Subtract 11x_{0} from both sides.

$$x_{0}^{2}+2x_{0}-\frac{151}{2}-11x_{0}=\frac{1}{2}x_{0}^{2}$$

Combine 2x_{0} and -11x_{0} to get -9x_{0}.

$$x_{0}^{2}-9x_{0}-\frac{151}{2}=\frac{1}{2}x_{0}^{2}$$

Subtract \frac{1}{2}x_{0}^{2} from both sides.

$$x_{0}^{2}-9x_{0}-\frac{151}{2}-\frac{1}{2}x_{0}^{2}=0$$

Combine x_{0}^{2} and -\frac{1}{2}x_{0}^{2} to get \frac{1}{2}x_{0}^{2}.

$$\frac{1}{2}x_{0}^{2}-9x_{0}-\frac{151}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, -9 for b, and -\frac{151}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x_{0}=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times \frac{1}{2}\left(-\frac{151}{2}\right)}}{2\times \frac{1}{2}}$$

Square -9.

$$x_{0}=\frac{-\left(-9\right)±\sqrt{81-4\times \frac{1}{2}\left(-\frac{151}{2}\right)}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$x_{0}=\frac{-\left(-9\right)±\sqrt{81-2\left(-\frac{151}{2}\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -\frac{151}{2}.

$$x_{0}=\frac{-\left(-9\right)±\sqrt{81+151}}{2\times \frac{1}{2}}$$

Add 81 to 151.

$$x_{0}=\frac{-\left(-9\right)±\sqrt{232}}{2\times \frac{1}{2}}$$

Take the square root of 232.

$$x_{0}=\frac{-\left(-9\right)±2\sqrt{58}}{2\times \frac{1}{2}}$$

The opposite of -9 is 9.

$$x_{0}=\frac{9±2\sqrt{58}}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$x_{0}=\frac{9±2\sqrt{58}}{1}$$

Now solve the equation x_{0}=\frac{9±2\sqrt{58}}{1} when ± is plus. Add 9 to 2\sqrt{58}.

$$x_{0}=\frac{2\sqrt{58}+9}{1}$$

Divide 9+2\sqrt{58} by 1.

$$x_{0}=2\sqrt{58}+9$$

Now solve the equation x_{0}=\frac{9±2\sqrt{58}}{1} when ± is minus. Subtract 2\sqrt{58} from 9.

$$x_{0}=\frac{9-2\sqrt{58}}{1}$$

Divide 9-2\sqrt{58} by 1.

$$x_{0}=9-2\sqrt{58}$$

The equation is now solved.

$$x_{0}=2\sqrt{58}+9 x_{0}=9-2\sqrt{58}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x_{0}+1\right)^{2}.

$$x_{0}^{2}+2x_{0}+1-4^{2}=\left(\frac{1\times 0-x_{0}-11}{\sqrt{2}}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$x_{0}^{2}+2x_{0}+1-16=\left(\frac{1\times 0-x_{0}-11}{\sqrt{2}}\right)^{2}$$

Subtract 16 from 1 to get -15.

$$x_{0}^{2}+2x_{0}-15=\left(\frac{1\times 0-x_{0}-11}{\sqrt{2}}\right)^{2}$$

Multiply 1 and 0 to get 0.

$$x_{0}^{2}+2x_{0}-15=\left(\frac{0-x_{0}-11}{\sqrt{2}}\right)^{2}$$

Subtract 11 from 0 to get -11.

$$x_{0}^{2}+2x_{0}-15=\left(\frac{-11-x_{0}}{\sqrt{2}}\right)^{2}$$

Rationalize the denominator of \frac{-11-x_{0}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$x_{0}^{2}+2x_{0}-15=\left(\frac{\left(-11-x_{0}\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\right)^{2}$$

The square of \sqrt{2} is 2.

$$x_{0}^{2}+2x_{0}-15=\left(\frac{\left(-11-x_{0}\right)\sqrt{2}}{2}\right)^{2}$$

To raise \frac{\left(-11-x_{0}\right)\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$x_{0}^{2}+2x_{0}-15=\frac{\left(\left(-11-x_{0}\right)\sqrt{2}\right)^{2}}{2^{2}}$$

Expand \left(\left(-11-x_{0}\right)\sqrt{2}\right)^{2}.

$$x_{0}^{2}+2x_{0}-15=\frac{\left(-11-x_{0}\right)^{2}\left(\sqrt{2}\right)^{2}}{2^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-11-x_{0}\right)^{2}.

$$x_{0}^{2}+2x_{0}-15=\frac{\left(121+22x_{0}+x_{0}^{2}\right)\left(\sqrt{2}\right)^{2}}{2^{2}}$$

The square of \sqrt{2} is 2.

$$x_{0}^{2}+2x_{0}-15=\frac{\left(121+22x_{0}+x_{0}^{2}\right)\times 2}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$x_{0}^{2}+2x_{0}-15=\frac{\left(121+22x_{0}+x_{0}^{2}\right)\times 2}{4}$$

Divide \left(121+22x_{0}+x_{0}^{2}\right)\times 2 by 4 to get \left(121+22x_{0}+x_{0}^{2}\right)\times \frac{1}{2}.

$$x_{0}^{2}+2x_{0}-15=\left(121+22x_{0}+x_{0}^{2}\right)\times \frac{1}{2}$$

Use the distributive property to multiply 121+22x_{0}+x_{0}^{2} by \frac{1}{2}.

$$x_{0}^{2}+2x_{0}-15=\frac{121}{2}+11x_{0}+\frac{1}{2}x_{0}^{2}$$

Subtract 11x_{0} from both sides.

$$x_{0}^{2}+2x_{0}-15-11x_{0}=\frac{121}{2}+\frac{1}{2}x_{0}^{2}$$

Combine 2x_{0} and -11x_{0} to get -9x_{0}.

$$x_{0}^{2}-9x_{0}-15=\frac{121}{2}+\frac{1}{2}x_{0}^{2}$$

Subtract \frac{1}{2}x_{0}^{2} from both sides.

$$x_{0}^{2}-9x_{0}-15-\frac{1}{2}x_{0}^{2}=\frac{121}{2}$$

Combine x_{0}^{2} and -\frac{1}{2}x_{0}^{2} to get \frac{1}{2}x_{0}^{2}.

$$\frac{1}{2}x_{0}^{2}-9x_{0}-15=\frac{121}{2}$$

Add 15 to both sides.

$$\frac{1}{2}x_{0}^{2}-9x_{0}=\frac{121}{2}+15$$

Add \frac{121}{2} and 15 to get \frac{151}{2}.

$$\frac{1}{2}x_{0}^{2}-9x_{0}=\frac{151}{2}$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}x_{0}^{2}-9x_{0}}{\frac{1}{2}}=\frac{\frac{151}{2}}{\frac{1}{2}}$$

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

$$x_{0}^{2}+\left(-\frac{9}{\frac{1}{2}}\right)x_{0}=\frac{\frac{151}{2}}{\frac{1}{2}}$$

Divide -9 by \frac{1}{2} by multiplying -9 by the reciprocal of \frac{1}{2}.

$$x_{0}^{2}-18x_{0}=\frac{\frac{151}{2}}{\frac{1}{2}}$$

Divide \frac{151}{2} by \frac{1}{2} by multiplying \frac{151}{2} by the reciprocal of \frac{1}{2}.

$$x_{0}^{2}-18x_{0}=151$$

Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x_{0}^{2}-18x_{0}+\left(-9\right)^{2}=151+\left(-9\right)^{2}$$

Square -9.

$$x_{0}^{2}-18x_{0}+81=151+81$$

Add 151 to 81.

$$x_{0}^{2}-18x_{0}+81=232$$

Factor x_{0}^{2}-18x_{0}+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x_{0}-9\right)^{2}=232$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x_{0}-9\right)^{2}}=\sqrt{232}$$

Simplify.

$$x_{0}-9=2\sqrt{58} x_{0}-9=-2\sqrt{58}$$

Add 9 to both sides of the equation.

$$x_{0}=2\sqrt{58}+9 x_{0}=9-2\sqrt{58}$$