Solve {left(6-14+x/8right)}^2+{left(4-42+x/8+14right)}^2=160

$${ \left(6- \frac{ 14+x }{ 8 } \right) }^{ 2 } + { \left(4- \frac{ 42+x }{ 8 } +14 \right) }^{ 2 } = 160$$

Solve for x

$$x=2\sqrt{991}+68\approx 130.960304955$$

$$x=68-2\sqrt{991}\approx 5.039695045$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6-\frac{14+x}{8}\right)^{2}.

$$36+12\left(-\frac{14+x}{8}\right)+\left(-\frac{14+x}{8}\right)^{2}+\left(4-\frac{42+x}{8}+14\right)^{2}=160$$

Express 12\left(-\frac{14+x}{8}\right) as a single fraction.

$$36+\frac{-12\left(14+x\right)}{8}+\left(-\frac{14+x}{8}\right)^{2}+\left(4-\frac{42+x}{8}+14\right)^{2}=160$$

Calculate -\frac{14+x}{8} to the power of 2 and get \left(\frac{14+x}{8}\right)^{2}.

$$36+\frac{-12\left(14+x\right)}{8}+\left(\frac{14+x}{8}\right)^{2}+\left(4-\frac{42+x}{8}+14\right)^{2}=160$$

Divide -12\left(14+x\right) by 8 to get -\frac{3}{2}\left(14+x\right).

$$36-\frac{3}{2}\left(14+x\right)+\left(\frac{14+x}{8}\right)^{2}+\left(4-\frac{42+x}{8}+14\right)^{2}=160$$

Use the distributive property to multiply -\frac{3}{2} by 14+x.

$$36-21-\frac{3}{2}x+\left(\frac{14+x}{8}\right)^{2}+\left(4-\frac{42+x}{8}+14\right)^{2}=160$$

Subtract 21 from 36 to get 15.

$$15-\frac{3}{2}x+\left(\frac{14+x}{8}\right)^{2}+\left(4-\frac{42+x}{8}+14\right)^{2}=160$$

To raise \frac{14+x}{8} to a power, raise both numerator and denominator to the power and then divide.

$$15-\frac{3}{2}x+\frac{\left(14+x\right)^{2}}{8^{2}}+\left(4-\frac{42+x}{8}+14\right)^{2}=160$$

Add 4 and 14 to get 18.

$$15-\frac{3}{2}x+\frac{\left(14+x\right)^{2}}{8^{2}}+\left(18-\frac{42+x}{8}\right)^{2}=160$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(18-\frac{42+x}{8}\right)^{2}.

$$15-\frac{3}{2}x+\frac{\left(14+x\right)^{2}}{8^{2}}+324+36\left(-\frac{42+x}{8}\right)+\left(-\frac{42+x}{8}\right)^{2}=160$$

Express 36\left(-\frac{42+x}{8}\right) as a single fraction.

$$15-\frac{3}{2}x+\frac{\left(14+x\right)^{2}}{8^{2}}+324+\frac{-36\left(42+x\right)}{8}+\left(-\frac{42+x}{8}\right)^{2}=160$$

Calculate -\frac{42+x}{8} to the power of 2 and get \left(\frac{42+x}{8}\right)^{2}.

$$15-\frac{3}{2}x+\frac{\left(14+x\right)^{2}}{8^{2}}+324+\frac{-36\left(42+x\right)}{8}+\left(\frac{42+x}{8}\right)^{2}=160$$

Add 15 and 324 to get 339.

$$339-\frac{3}{2}x+\frac{\left(14+x\right)^{2}}{8^{2}}+\frac{-36\left(42+x\right)}{8}+\left(\frac{42+x}{8}\right)^{2}=160$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(14+x\right)^{2}.

$$339-\frac{3}{2}x+\frac{196+28x+x^{2}}{8^{2}}+\frac{-36\left(42+x\right)}{8}+\left(\frac{42+x}{8}\right)^{2}=160$$

Calculate 8 to the power of 2 and get 64.

$$339-\frac{3}{2}x+\frac{196+28x+x^{2}}{64}+\frac{-36\left(42+x\right)}{8}+\left(\frac{42+x}{8}\right)^{2}=160$$

Divide -36\left(42+x\right) by 8 to get -\frac{9}{2}\left(42+x\right).

$$339-\frac{3}{2}x+\frac{196+28x+x^{2}}{64}-\frac{9}{2}\left(42+x\right)+\left(\frac{42+x}{8}\right)^{2}=160$$

Use the distributive property to multiply -\frac{9}{2} by 42+x.

$$339-\frac{3}{2}x+\frac{196+28x+x^{2}}{64}-189-\frac{9}{2}x+\left(\frac{42+x}{8}\right)^{2}=160$$

Subtract 189 from 339 to get 150.

$$150-\frac{3}{2}x+\frac{196+28x+x^{2}}{64}-\frac{9}{2}x+\left(\frac{42+x}{8}\right)^{2}=160$$

Combine -\frac{3}{2}x and -\frac{9}{2}x to get -6x.

$$150-6x+\frac{196+28x+x^{2}}{64}+\left(\frac{42+x}{8}\right)^{2}=160$$

To raise \frac{42+x}{8} to a power, raise both numerator and denominator to the power and then divide.

$$150-6x+\frac{196+28x+x^{2}}{64}+\frac{\left(42+x\right)^{2}}{8^{2}}=160$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 150-6x times \frac{8^{2}}{8^{2}}.

$$\frac{\left(150-6x\right)\times 8^{2}}{8^{2}}+\frac{196+28x+x^{2}}{64}+\frac{\left(42+x\right)^{2}}{8^{2}}=160$$

Since \frac{\left(150-6x\right)\times 8^{2}}{8^{2}} and \frac{\left(42+x\right)^{2}}{8^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(150-6x\right)\times 8^{2}+\left(42+x\right)^{2}}{8^{2}}+\frac{196+28x+x^{2}}{64}=160$$

Do the multiplications in \left(150-6x\right)\times 8^{2}+\left(42+x\right)^{2}.

$$\frac{9600-384x+1764+84x+x^{2}}{8^{2}}+\frac{196+28x+x^{2}}{64}=160$$

Combine like terms in 9600-384x+1764+84x+x^{2}.

$$\frac{11364-300x+x^{2}}{8^{2}}+\frac{196+28x+x^{2}}{64}=160$$

To add or subtract expressions, expand them to make their denominators the same. Expand 8^{2}.

$$\frac{11364-300x+x^{2}}{64}+\frac{196+28x+x^{2}}{64}=160$$

Since \frac{11364-300x+x^{2}}{64} and \frac{196+28x+x^{2}}{64} have the same denominator, add them by adding their numerators.

$$\frac{11364-300x+x^{2}+196+28x+x^{2}}{64}=160$$

Combine like terms in 11364-300x+x^{2}+196+28x+x^{2}.

$$\frac{11560-272x+2x^{2}}{64}=160$$

Divide each term of 11560-272x+2x^{2} by 64 to get \frac{1445}{8}-\frac{17}{4}x+\frac{1}{32}x^{2}.

$$\frac{1445}{8}-\frac{17}{4}x+\frac{1}{32}x^{2}=160$$

Subtract 160 from both sides.

$$\frac{1445}{8}-\frac{17}{4}x+\frac{1}{32}x^{2}-160=0$$

Subtract 160 from \frac{1445}{8} to get \frac{165}{8}.

$$\frac{165}{8}-\frac{17}{4}x+\frac{1}{32}x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$\frac{1}{32}x^{2}-\frac{17}{4}x+\frac{165}{8}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{32} for a, -\frac{17}{4} for b, and \frac{165}{8} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{17}{4}\right)±\sqrt{\left(-\frac{17}{4}\right)^{2}-4\times \frac{1}{32}\times \frac{165}{8}}}{2\times \frac{1}{32}}$$

Square -\frac{17}{4} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{17}{4}\right)±\sqrt{\frac{289}{16}-4\times \frac{1}{32}\times \frac{165}{8}}}{2\times \frac{1}{32}}$$

Multiply -4 times \frac{1}{32}.

$$x=\frac{-\left(-\frac{17}{4}\right)±\sqrt{\frac{289}{16}-\frac{1}{8}\times \frac{165}{8}}}{2\times \frac{1}{32}}$$

Multiply -\frac{1}{8} times \frac{165}{8} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{17}{4}\right)±\sqrt{\frac{289}{16}-\frac{165}{64}}}{2\times \frac{1}{32}}$$

Add \frac{289}{16} to -\frac{165}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{17}{4}\right)±\sqrt{\frac{991}{64}}}{2\times \frac{1}{32}}$$

Take the square root of \frac{991}{64}.

$$x=\frac{-\left(-\frac{17}{4}\right)±\frac{\sqrt{991}}{8}}{2\times \frac{1}{32}}$$

The opposite of -\frac{17}{4} is \frac{17}{4}.

$$x=\frac{\frac{17}{4}±\frac{\sqrt{991}}{8}}{2\times \frac{1}{32}}$$

Multiply 2 times \frac{1}{32}.

$$x=\frac{\frac{17}{4}±\frac{\sqrt{991}}{8}}{\frac{1}{16}}$$

Now solve the equation x=\frac{\frac{17}{4}±\frac{\sqrt{991}}{8}}{\frac{1}{16}} when ± is plus. Add \frac{17}{4} to \frac{\sqrt{991}}{8}.

$$x=\frac{\frac{\sqrt{991}}{8}+\frac{17}{4}}{\frac{1}{16}}$$

Divide \frac{17}{4}+\frac{\sqrt{991}}{8} by \frac{1}{16} by multiplying \frac{17}{4}+\frac{\sqrt{991}}{8} by the reciprocal of \frac{1}{16}.

$$x=2\sqrt{991}+68$$

Now solve the equation x=\frac{\frac{17}{4}±\frac{\sqrt{991}}{8}}{\frac{1}{16}} when ± is minus. Subtract \frac{\sqrt{991}}{8} from \frac{17}{4}.

$$x=\frac{-\frac{\sqrt{991}}{8}+\frac{17}{4}}{\frac{1}{16}}$$

Divide \frac{17}{4}-\frac{\sqrt{991}}{8} by \frac{1}{16} by multiplying \frac{17}{4}-\frac{\sqrt{991}}{8} by the reciprocal of \frac{1}{16}.

$$x=68-2\sqrt{991}$$

The equation is now solved.

$$x=2\sqrt{991}+68 x=68-2\sqrt{991}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6-\frac{14+x}{8}\right)^{2}.

$$36+12\left(-\frac{14+x}{8}\right)+\left(-\frac{14+x}{8}\right)^{2}+\left(4-\frac{42+x}{8}+14\right)^{2}=160$$

Express 12\left(-\frac{14+x}{8}\right) as a single fraction.

$$36+\frac{-12\left(14+x\right)}{8}+\left(-\frac{14+x}{8}\right)^{2}+\left(4-\frac{42+x}{8}+14\right)^{2}=160$$

Calculate -\frac{14+x}{8} to the power of 2 and get \left(\frac{14+x}{8}\right)^{2}.

$$36+\frac{-12\left(14+x\right)}{8}+\left(\frac{14+x}{8}\right)^{2}+\left(4-\frac{42+x}{8}+14\right)^{2}=160$$

Divide -12\left(14+x\right) by 8 to get -\frac{3}{2}\left(14+x\right).

$$36-\frac{3}{2}\left(14+x\right)+\left(\frac{14+x}{8}\right)^{2}+\left(4-\frac{42+x}{8}+14\right)^{2}=160$$

Use the distributive property to multiply -\frac{3}{2} by 14+x.

$$36-21-\frac{3}{2}x+\left(\frac{14+x}{8}\right)^{2}+\left(4-\frac{42+x}{8}+14\right)^{2}=160$$

Subtract 21 from 36 to get 15.

$$15-\frac{3}{2}x+\left(\frac{14+x}{8}\right)^{2}+\left(4-\frac{42+x}{8}+14\right)^{2}=160$$

To raise \frac{14+x}{8} to a power, raise both numerator and denominator to the power and then divide.

$$15-\frac{3}{2}x+\frac{\left(14+x\right)^{2}}{8^{2}}+\left(4-\frac{42+x}{8}+14\right)^{2}=160$$

Add 4 and 14 to get 18.

$$15-\frac{3}{2}x+\frac{\left(14+x\right)^{2}}{8^{2}}+\left(18-\frac{42+x}{8}\right)^{2}=160$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(18-\frac{42+x}{8}\right)^{2}.

$$15-\frac{3}{2}x+\frac{\left(14+x\right)^{2}}{8^{2}}+324+36\left(-\frac{42+x}{8}\right)+\left(-\frac{42+x}{8}\right)^{2}=160$$

Express 36\left(-\frac{42+x}{8}\right) as a single fraction.

$$15-\frac{3}{2}x+\frac{\left(14+x\right)^{2}}{8^{2}}+324+\frac{-36\left(42+x\right)}{8}+\left(-\frac{42+x}{8}\right)^{2}=160$$

Calculate -\frac{42+x}{8} to the power of 2 and get \left(\frac{42+x}{8}\right)^{2}.

$$15-\frac{3}{2}x+\frac{\left(14+x\right)^{2}}{8^{2}}+324+\frac{-36\left(42+x\right)}{8}+\left(\frac{42+x}{8}\right)^{2}=160$$

Add 15 and 324 to get 339.

$$339-\frac{3}{2}x+\frac{\left(14+x\right)^{2}}{8^{2}}+\frac{-36\left(42+x\right)}{8}+\left(\frac{42+x}{8}\right)^{2}=160$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(14+x\right)^{2}.

$$339-\frac{3}{2}x+\frac{196+28x+x^{2}}{8^{2}}+\frac{-36\left(42+x\right)}{8}+\left(\frac{42+x}{8}\right)^{2}=160$$

Calculate 8 to the power of 2 and get 64.

$$339-\frac{3}{2}x+\frac{196+28x+x^{2}}{64}+\frac{-36\left(42+x\right)}{8}+\left(\frac{42+x}{8}\right)^{2}=160$$

Divide -36\left(42+x\right) by 8 to get -\frac{9}{2}\left(42+x\right).

$$339-\frac{3}{2}x+\frac{196+28x+x^{2}}{64}-\frac{9}{2}\left(42+x\right)+\left(\frac{42+x}{8}\right)^{2}=160$$

Use the distributive property to multiply -\frac{9}{2} by 42+x.

$$339-\frac{3}{2}x+\frac{196+28x+x^{2}}{64}-189-\frac{9}{2}x+\left(\frac{42+x}{8}\right)^{2}=160$$

Subtract 189 from 339 to get 150.

$$150-\frac{3}{2}x+\frac{196+28x+x^{2}}{64}-\frac{9}{2}x+\left(\frac{42+x}{8}\right)^{2}=160$$

Combine -\frac{3}{2}x and -\frac{9}{2}x to get -6x.

$$150-6x+\frac{196+28x+x^{2}}{64}+\left(\frac{42+x}{8}\right)^{2}=160$$

To raise \frac{42+x}{8} to a power, raise both numerator and denominator to the power and then divide.

$$150-6x+\frac{196+28x+x^{2}}{64}+\frac{\left(42+x\right)^{2}}{8^{2}}=160$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 150-6x times \frac{8^{2}}{8^{2}}.

$$\frac{\left(150-6x\right)\times 8^{2}}{8^{2}}+\frac{196+28x+x^{2}}{64}+\frac{\left(42+x\right)^{2}}{8^{2}}=160$$

Since \frac{\left(150-6x\right)\times 8^{2}}{8^{2}} and \frac{\left(42+x\right)^{2}}{8^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(150-6x\right)\times 8^{2}+\left(42+x\right)^{2}}{8^{2}}+\frac{196+28x+x^{2}}{64}=160$$

Do the multiplications in \left(150-6x\right)\times 8^{2}+\left(42+x\right)^{2}.

$$\frac{9600-384x+1764+84x+x^{2}}{8^{2}}+\frac{196+28x+x^{2}}{64}=160$$

Combine like terms in 9600-384x+1764+84x+x^{2}.

$$\frac{11364-300x+x^{2}}{8^{2}}+\frac{196+28x+x^{2}}{64}=160$$

To add or subtract expressions, expand them to make their denominators the same. Expand 8^{2}.

$$\frac{11364-300x+x^{2}}{64}+\frac{196+28x+x^{2}}{64}=160$$

Since \frac{11364-300x+x^{2}}{64} and \frac{196+28x+x^{2}}{64} have the same denominator, add them by adding their numerators.

$$\frac{11364-300x+x^{2}+196+28x+x^{2}}{64}=160$$

Combine like terms in 11364-300x+x^{2}+196+28x+x^{2}.

$$\frac{11560-272x+2x^{2}}{64}=160$$

Divide each term of 11560-272x+2x^{2} by 64 to get \frac{1445}{8}-\frac{17}{4}x+\frac{1}{32}x^{2}.

$$\frac{1445}{8}-\frac{17}{4}x+\frac{1}{32}x^{2}=160$$

Subtract \frac{1445}{8} from both sides.

$$-\frac{17}{4}x+\frac{1}{32}x^{2}=160-\frac{1445}{8}$$

Subtract \frac{1445}{8} from 160 to get -\frac{165}{8}.

$$-\frac{17}{4}x+\frac{1}{32}x^{2}=-\frac{165}{8}$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$\frac{1}{32}x^{2}-\frac{17}{4}x=-\frac{165}{8}$$

Multiply both sides by 32.

$$\frac{\frac{1}{32}x^{2}-\frac{17}{4}x}{\frac{1}{32}}=-\frac{\frac{165}{8}}{\frac{1}{32}}$$

Dividing by \frac{1}{32} undoes the multiplication by \frac{1}{32}.

$$x^{2}+\left(-\frac{\frac{17}{4}}{\frac{1}{32}}\right)x=-\frac{\frac{165}{8}}{\frac{1}{32}}$$

Divide -\frac{17}{4} by \frac{1}{32} by multiplying -\frac{17}{4} by the reciprocal of \frac{1}{32}.

$$x^{2}-136x=-\frac{\frac{165}{8}}{\frac{1}{32}}$$

Divide -\frac{165}{8} by \frac{1}{32} by multiplying -\frac{165}{8} by the reciprocal of \frac{1}{32}.

$$x^{2}-136x=-660$$

Divide -136, the coefficient of the x term, by 2 to get -68. Then add the square of -68 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-136x+\left(-68\right)^{2}=-660+\left(-68\right)^{2}$$

Square -68.

$$x^{2}-136x+4624=-660+4624$$

Add -660 to 4624.

$$x^{2}-136x+4624=3964$$

Factor x^{2}-136x+4624. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-68\right)^{2}=3964$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-68\right)^{2}}=\sqrt{3964}$$

Simplify.

$$x-68=2\sqrt{991} x-68=-2\sqrt{991}$$

Add 68 to both sides of the equation.

$$x=2\sqrt{991}+68 x=68-2\sqrt{991}$$