Solve {left(0.1sqrt{2}x-30sqrt{2}right)}^2+{left(210sqrt{2}-0.1sqrt{2}xright)}^2=left(60+10xright)^2

$${ \left(0.1 \sqrt{ 2 } x-30 \sqrt{ 2 } \right) }^{ 2 } + { \left(210 \sqrt{ 2 } -0.1 \sqrt{ 2 } x \right) }^{ 2 } = { \left(60+10x \right) }^{ 2 }$$

Solve for x

$$x = \frac{600 \sqrt{1747} - 5400}{833} \approx 23.623382296$$

$$x=\frac{-600\sqrt{1747}-5400}{833}\approx -36.588568371$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(0.1\sqrt{2}x-30\sqrt{2}\right)^{2}.

$$0.01\left(\sqrt{2}\right)^{2}x^{2}-6\sqrt{2}x\sqrt{2}+900\left(\sqrt{2}\right)^{2}+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=\left(60+10x\right)^{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$0.01\left(\sqrt{2}\right)^{2}x^{2}-6\times 2x+900\left(\sqrt{2}\right)^{2}+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=\left(60+10x\right)^{2}$$

The square of \sqrt{2} is 2.

$$0.01\times 2x^{2}-6\times 2x+900\left(\sqrt{2}\right)^{2}+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=\left(60+10x\right)^{2}$$

Multiply 0.01 and 2 to get 0.02.

$$0.02x^{2}-6\times 2x+900\left(\sqrt{2}\right)^{2}+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=\left(60+10x\right)^{2}$$

Multiply -6 and 2 to get -12.

$$0.02x^{2}-12x+900\left(\sqrt{2}\right)^{2}+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=\left(60+10x\right)^{2}$$

The square of \sqrt{2} is 2.

$$0.02x^{2}-12x+900\times 2+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=\left(60+10x\right)^{2}$$

Multiply 900 and 2 to get 1800.

$$0.02x^{2}-12x+1800+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=\left(60+10x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(60+10x\right)^{2}.

$$0.02x^{2}-12x+1800+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=3600+1200x+100x^{2}$$

Subtract 3600 from both sides.

$$0.02x^{2}-12x+1800+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}-3600=1200x+100x^{2}$$

Subtract 3600 from 1800 to get -1800.

$$0.02x^{2}-12x-1800+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=1200x+100x^{2}$$

Subtract 1200x from both sides.

$$0.02x^{2}-12x-1800+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}-1200x=100x^{2}$$

Combine -12x and -1200x to get -1212x.

$$0.02x^{2}-1212x-1800+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=100x^{2}$$

Subtract 100x^{2} from both sides.

$$0.02x^{2}-1212x-1800+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}-100x^{2}=0$$

Combine 0.02x^{2} and -100x^{2} to get -99.98x^{2}.

$$-99.98x^{2}-1212x-1800+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}.

$$-99.98x^{2}-1212x-1800+44100\left(\sqrt{2}\right)^{2}-42\sqrt{2}\sqrt{2}x+0.01\left(\sqrt{2}\right)^{2}x^{2}=0$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$-99.98x^{2}-1212x-1800+44100\left(\sqrt{2}\right)^{2}-42\times 2x+0.01\left(\sqrt{2}\right)^{2}x^{2}=0$$

The square of \sqrt{2} is 2.

$$-99.98x^{2}-1212x-1800+44100\times 2-42\times 2x+0.01\left(\sqrt{2}\right)^{2}x^{2}=0$$

Multiply 44100 and 2 to get 88200.

$$-99.98x^{2}-1212x-1800+88200-42\times 2x+0.01\left(\sqrt{2}\right)^{2}x^{2}=0$$

Multiply -42 and 2 to get -84.

$$-99.98x^{2}-1212x-1800+88200-84x+0.01\left(\sqrt{2}\right)^{2}x^{2}=0$$

The square of \sqrt{2} is 2.

$$-99.98x^{2}-1212x-1800+88200-84x+0.01\times 2x^{2}=0$$

Multiply 0.01 and 2 to get 0.02.

$$-99.98x^{2}-1212x-1800+88200-84x+0.02x^{2}=0$$

Add -1800 and 88200 to get 86400.

$$-99.98x^{2}-1212x+86400-84x+0.02x^{2}=0$$

Combine -1212x and -84x to get -1296x.

$$-99.98x^{2}-1296x+86400+0.02x^{2}=0$$

Combine -99.98x^{2} and 0.02x^{2} to get -99.96x^{2}.

$$-99.96x^{2}-1296x+86400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -99.96 for a, -1296 for b, and 86400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1296\right)±\sqrt{\left(-1296\right)^{2}-4\left(-99.96\right)\times 86400}}{2\left(-99.96\right)}$$

Square -1296.

$$x=\frac{-\left(-1296\right)±\sqrt{1679616-4\left(-99.96\right)\times 86400}}{2\left(-99.96\right)}$$

Multiply -4 times -99.96.

$$x=\frac{-\left(-1296\right)±\sqrt{1679616+399.84\times 86400}}{2\left(-99.96\right)}$$

Multiply 399.84 times 86400.

$$x=\frac{-\left(-1296\right)±\sqrt{1679616+34546176}}{2\left(-99.96\right)}$$

Add 1679616 to 34546176.

$$x=\frac{-\left(-1296\right)±\sqrt{36225792}}{2\left(-99.96\right)}$$

Take the square root of 36225792.

$$x=\frac{-\left(-1296\right)±144\sqrt{1747}}{2\left(-99.96\right)}$$

The opposite of -1296 is 1296.

$$x=\frac{1296±144\sqrt{1747}}{2\left(-99.96\right)}$$

Multiply 2 times -99.96.

$$x=\frac{1296±144\sqrt{1747}}{-199.92}$$

Now solve the equation x=\frac{1296±144\sqrt{1747}}{-199.92} when ± is plus. Add 1296 to 144\sqrt{1747}.

$$x=\frac{144\sqrt{1747}+1296}{-199.92}$$

Divide 1296+144\sqrt{1747} by -199.92 by multiplying 1296+144\sqrt{1747} by the reciprocal of -199.92.

$$x=\frac{-600\sqrt{1747}-5400}{833}$$

Now solve the equation x=\frac{1296±144\sqrt{1747}}{-199.92} when ± is minus. Subtract 144\sqrt{1747} from 1296.

$$x=\frac{1296-144\sqrt{1747}}{-199.92}$$

Divide 1296-144\sqrt{1747} by -199.92 by multiplying 1296-144\sqrt{1747} by the reciprocal of -199.92.

$$x=\frac{600\sqrt{1747}-5400}{833}$$

The equation is now solved.

$$x=\frac{-600\sqrt{1747}-5400}{833} x=\frac{600\sqrt{1747}-5400}{833}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(0.1\sqrt{2}x-30\sqrt{2}\right)^{2}.

$$0.01\left(\sqrt{2}\right)^{2}x^{2}-6\sqrt{2}x\sqrt{2}+900\left(\sqrt{2}\right)^{2}+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=\left(60+10x\right)^{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$0.01\left(\sqrt{2}\right)^{2}x^{2}-6\times 2x+900\left(\sqrt{2}\right)^{2}+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=\left(60+10x\right)^{2}$$

The square of \sqrt{2} is 2.

$$0.01\times 2x^{2}-6\times 2x+900\left(\sqrt{2}\right)^{2}+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=\left(60+10x\right)^{2}$$

Multiply 0.01 and 2 to get 0.02.

$$0.02x^{2}-6\times 2x+900\left(\sqrt{2}\right)^{2}+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=\left(60+10x\right)^{2}$$

Multiply -6 and 2 to get -12.

$$0.02x^{2}-12x+900\left(\sqrt{2}\right)^{2}+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=\left(60+10x\right)^{2}$$

The square of \sqrt{2} is 2.

$$0.02x^{2}-12x+900\times 2+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=\left(60+10x\right)^{2}$$

Multiply 900 and 2 to get 1800.

$$0.02x^{2}-12x+1800+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=\left(60+10x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(60+10x\right)^{2}.

$$0.02x^{2}-12x+1800+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=3600+1200x+100x^{2}$$

Subtract 1200x from both sides.

$$0.02x^{2}-12x+1800+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}-1200x=3600+100x^{2}$$

Combine -12x and -1200x to get -1212x.

$$0.02x^{2}-1212x+1800+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=3600+100x^{2}$$

Subtract 100x^{2} from both sides.

$$0.02x^{2}-1212x+1800+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}-100x^{2}=3600$$

Combine 0.02x^{2} and -100x^{2} to get -99.98x^{2}.

$$-99.98x^{2}-1212x+1800+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=3600$$

Subtract 1800 from both sides.

$$-99.98x^{2}-1212x+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=3600-1800$$

Subtract 1800 from 3600 to get 1800.

$$-99.98x^{2}-1212x+\left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}=1800$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(210\sqrt{2}-0.1\sqrt{2}x\right)^{2}.

$$-99.98x^{2}-1212x+44100\left(\sqrt{2}\right)^{2}-42\sqrt{2}\sqrt{2}x+0.01\left(\sqrt{2}\right)^{2}x^{2}=1800$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$-99.98x^{2}-1212x+44100\left(\sqrt{2}\right)^{2}-42\times 2x+0.01\left(\sqrt{2}\right)^{2}x^{2}=1800$$

The square of \sqrt{2} is 2.

$$-99.98x^{2}-1212x+44100\times 2-42\times 2x+0.01\left(\sqrt{2}\right)^{2}x^{2}=1800$$

Multiply 44100 and 2 to get 88200.

$$-99.98x^{2}-1212x+88200-42\times 2x+0.01\left(\sqrt{2}\right)^{2}x^{2}=1800$$

Multiply -42 and 2 to get -84.

$$-99.98x^{2}-1212x+88200-84x+0.01\left(\sqrt{2}\right)^{2}x^{2}=1800$$

The square of \sqrt{2} is 2.

$$-99.98x^{2}-1212x+88200-84x+0.01\times 2x^{2}=1800$$

Multiply 0.01 and 2 to get 0.02.

$$-99.98x^{2}-1212x+88200-84x+0.02x^{2}=1800$$

Combine -1212x and -84x to get -1296x.

$$-99.98x^{2}-1296x+88200+0.02x^{2}=1800$$

Combine -99.98x^{2} and 0.02x^{2} to get -99.96x^{2}.

$$-99.96x^{2}-1296x+88200=1800$$

Subtract 88200 from both sides.

$$-99.96x^{2}-1296x=1800-88200$$

Subtract 88200 from 1800 to get -86400.

$$-99.96x^{2}-1296x=-86400$$

Divide both sides of the equation by -99.96, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-99.96x^{2}-1296x}{-99.96}=-\frac{86400}{-99.96}$$

Dividing by -99.96 undoes the multiplication by -99.96.

$$x^{2}+\left(-\frac{1296}{-99.96}\right)x=-\frac{86400}{-99.96}$$

Divide -1296 by -99.96 by multiplying -1296 by the reciprocal of -99.96.

$$x^{2}+\frac{10800}{833}x=-\frac{86400}{-99.96}$$

Divide -86400 by -99.96 by multiplying -86400 by the reciprocal of -99.96.

$$x^{2}+\frac{10800}{833}x=\frac{720000}{833}$$

Divide \frac{10800}{833}, the coefficient of the x term, by 2 to get \frac{5400}{833}. Then add the square of \frac{5400}{833} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{10800}{833}x+\frac{5400}{833}^{2}=\frac{720000}{833}+\frac{5400}{833}^{2}$$

Square \frac{5400}{833} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{10800}{833}x+\frac{29160000}{693889}=\frac{720000}{833}+\frac{29160000}{693889}$$

Add \frac{720000}{833} to \frac{29160000}{693889} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{10800}{833}x+\frac{29160000}{693889}=\frac{628920000}{693889}$$

Factor x^{2}+\frac{10800}{833}x+\frac{29160000}{693889}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5400}{833}\right)^{2}=\frac{628920000}{693889}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5400}{833}\right)^{2}}=\sqrt{\frac{628920000}{693889}}$$

Simplify.

$$x+\frac{5400}{833}=\frac{600\sqrt{1747}}{833} x+\frac{5400}{833}=-\frac{600\sqrt{1747}}{833}$$

Subtract \frac{5400}{833} from both sides of the equation.

$$x=\frac{600\sqrt{1747}-5400}{833} x=\frac{-600\sqrt{1747}-5400}{833}$$