Solve 45={left(1-x/24right)}^2

$$45= { \left(1- \frac{ x }{ 24 } \right) }^{ 2 }$$

Solve for x

$$x=24-72\sqrt{5}\approx -136.99689438$$

$$x=72\sqrt{5}+24\approx 184.99689438$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1-\frac{x}{24}\right)^{2}.

$$45=1+2\left(-\frac{x}{24}\right)+\left(-\frac{x}{24}\right)^{2}$$

Cancel out 24, the greatest common factor in 2 and 24.

$$45=1+\frac{x}{-12}+\left(-\frac{x}{24}\right)^{2}$$

Calculate -\frac{x}{24} to the power of 2 and get \left(\frac{x}{24}\right)^{2}.

$$45=1+\frac{x}{-12}+\left(\frac{x}{24}\right)^{2}$$

To raise \frac{x}{24} to a power, raise both numerator and denominator to the power and then divide.

$$45=1+\frac{x}{-12}+\frac{x^{2}}{24^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{24^{2}}{24^{2}}.

$$45=\frac{24^{2}}{24^{2}}+\frac{x}{-12}+\frac{x^{2}}{24^{2}}$$

Since \frac{24^{2}}{24^{2}} and \frac{x^{2}}{24^{2}} have the same denominator, add them by adding their numerators.

$$45=\frac{24^{2}+x^{2}}{24^{2}}+\frac{x}{-12}$$

Combine like terms in 24^{2}+x^{2}.

$$45=\frac{576+x^{2}}{24^{2}}+\frac{x}{-12}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{576+x^{2}}{24^{2}}+\frac{x}{-12}=45$$

Subtract 45 from both sides.

$$\frac{576+x^{2}}{24^{2}}+\frac{x}{-12}-45=0$$

Calculate 24 to the power of 2 and get 576.

$$\frac{576+x^{2}}{576}+\frac{x}{-12}-45=0$$

Divide each term of 576+x^{2} by 576 to get 1+\frac{1}{576}x^{2}.

$$1+\frac{1}{576}x^{2}+\frac{x}{-12}-45=0$$

Subtract 45 from 1 to get -44.

$$-44+\frac{1}{576}x^{2}+\frac{x}{-12}=0$$

Multiply both sides of the equation by 576, the least common multiple of 576,-12.

$$-25344+1x^{2}-48x=0$$

Reorder the terms.

$$x^{2}-48x-25344=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -48 for b, and -25344 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\left(-25344\right)}}{2}$$

Square -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\left(-25344\right)}}{2}$$

Multiply -4 times -25344.

$$x=\frac{-\left(-48\right)±\sqrt{2304+101376}}{2}$$

Add 2304 to 101376.

$$x=\frac{-\left(-48\right)±\sqrt{103680}}{2}$$

Take the square root of 103680.

$$x=\frac{-\left(-48\right)±144\sqrt{5}}{2}$$

The opposite of -48 is 48.

$$x=\frac{48±144\sqrt{5}}{2}$$

Now solve the equation x=\frac{48±144\sqrt{5}}{2} when ± is plus. Add 48 to 144\sqrt{5}.

$$x=\frac{144\sqrt{5}+48}{2}$$

Divide 48+144\sqrt{5} by 2.

$$x=72\sqrt{5}+24$$

Now solve the equation x=\frac{48±144\sqrt{5}}{2} when ± is minus. Subtract 144\sqrt{5} from 48.

$$x=\frac{48-144\sqrt{5}}{2}$$

Divide 48-144\sqrt{5} by 2.

$$x=24-72\sqrt{5}$$

The equation is now solved.

$$x=72\sqrt{5}+24 x=24-72\sqrt{5}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1-\frac{x}{24}\right)^{2}.

$$45=1+2\left(-\frac{x}{24}\right)+\left(-\frac{x}{24}\right)^{2}$$

Cancel out 24, the greatest common factor in 2 and 24.

$$45=1+\frac{x}{-12}+\left(-\frac{x}{24}\right)^{2}$$

Calculate -\frac{x}{24} to the power of 2 and get \left(\frac{x}{24}\right)^{2}.

$$45=1+\frac{x}{-12}+\left(\frac{x}{24}\right)^{2}$$

To raise \frac{x}{24} to a power, raise both numerator and denominator to the power and then divide.

$$45=1+\frac{x}{-12}+\frac{x^{2}}{24^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{24^{2}}{24^{2}}.

$$45=\frac{24^{2}}{24^{2}}+\frac{x}{-12}+\frac{x^{2}}{24^{2}}$$

Since \frac{24^{2}}{24^{2}} and \frac{x^{2}}{24^{2}} have the same denominator, add them by adding their numerators.

$$45=\frac{24^{2}+x^{2}}{24^{2}}+\frac{x}{-12}$$

Combine like terms in 24^{2}+x^{2}.

$$45=\frac{576+x^{2}}{24^{2}}+\frac{x}{-12}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{576+x^{2}}{24^{2}}+\frac{x}{-12}=45$$

Calculate 24 to the power of 2 and get 576.

$$\frac{576+x^{2}}{576}+\frac{x}{-12}=45$$

Divide each term of 576+x^{2} by 576 to get 1+\frac{1}{576}x^{2}.

$$1+\frac{1}{576}x^{2}+\frac{x}{-12}=45$$

Subtract 1 from both sides.

$$\frac{1}{576}x^{2}+\frac{x}{-12}=45-1$$

Subtract 1 from 45 to get 44.

$$\frac{1}{576}x^{2}+\frac{x}{-12}=44$$

Multiply both sides of the equation by 576, the least common multiple of 576,-12.

$$1x^{2}-48x=25344$$

Reorder the terms.

$$x^{2}-48x=25344$$

Divide -48, the coefficient of the x term, by 2 to get -24. Then add the square of -24 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-48x+\left(-24\right)^{2}=25344+\left(-24\right)^{2}$$

Square -24.

$$x^{2}-48x+576=25344+576$$

Add 25344 to 576.

$$x^{2}-48x+576=25920$$

Factor x^{2}-48x+576. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-24\right)^{2}=25920$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-24\right)^{2}}=\sqrt{25920}$$

Simplify.

$$x-24=72\sqrt{5} x-24=-72\sqrt{5}$$

Add 24 to both sides of the equation.

$$x=72\sqrt{5}+24 x=24-72\sqrt{5}$$