Solve left(100-xright)^2+left(41+0.41xright)^2=10000

$${ \left(100-x \right) }^{ 2 } + { \left(41+0.41x \right) }^{ 2 } =10000$$

Solve for x

$$x = \frac{10000 \sqrt{4957} + 831900}{11681} \approx 131.492137365$$

$$x = \frac{831900 - 10000 \sqrt{4957}}{11681} \approx 10.944297871$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(100-x\right)^{2}.

$$10000-200x+x^{2}+\left(41+0.41x\right)^{2}=10000$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(41+0.41x\right)^{2}.

$$10000-200x+x^{2}+1681+33.62x+0.1681x^{2}=10000$$

Add 10000 and 1681 to get 11681.

$$11681-200x+x^{2}+33.62x+0.1681x^{2}=10000$$

Combine -200x and 33.62x to get -166.38x.

$$11681-166.38x+x^{2}+0.1681x^{2}=10000$$

Combine x^{2} and 0.1681x^{2} to get 1.1681x^{2}.

$$11681-166.38x+1.1681x^{2}=10000$$

Subtract 10000 from both sides.

$$11681-166.38x+1.1681x^{2}-10000=0$$

Subtract 10000 from 11681 to get 1681.

$$1681-166.38x+1.1681x^{2}=0$$

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.

$$1.1681x^{2}-166.38x+1681=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.1681 for a, -166.38 for b, and 1681 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-166.38\right)±\sqrt{\left(-166.38\right)^{2}-4\times 1.1681\times 1681}}{2\times 1.1681}$$

Square -166.38 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-166.38\right)±\sqrt{27682.3044-4\times 1.1681\times 1681}}{2\times 1.1681}$$

Multiply -4 times 1.1681.

$$x=\frac{-\left(-166.38\right)±\sqrt{27682.3044-4.6724\times 1681}}{2\times 1.1681}$$

Multiply -4.6724 times 1681.

$$x=\frac{-\left(-166.38\right)±\sqrt{\frac{69205761-19635761}{2500}}}{2\times 1.1681}$$

Add 27682.3044 to -7854.3044 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-166.38\right)±\sqrt{19828}}{2\times 1.1681}$$

Take the square root of 19828.

$$x=\frac{-\left(-166.38\right)±2\sqrt{4957}}{2\times 1.1681}$$

The opposite of -166.38 is 166.38.

$$x=\frac{166.38±2\sqrt{4957}}{2\times 1.1681}$$

Multiply 2 times 1.1681.

$$x=\frac{166.38±2\sqrt{4957}}{2.3362}$$

Now solve the equation x=\frac{166.38±2\sqrt{4957}}{2.3362} when ± is plus. Add 166.38 to 2\sqrt{4957}.

$$x=\frac{2\sqrt{4957}+166.38}{2.3362}$$

Divide 166.38+2\sqrt{4957} by 2.3362 by multiplying 166.38+2\sqrt{4957} by the reciprocal of 2.3362.

$$x=\frac{10000\sqrt{4957}+831900}{11681}$$

Now solve the equation x=\frac{166.38±2\sqrt{4957}}{2.3362} when ± is minus. Subtract 2\sqrt{4957} from 166.38.

$$x=\frac{166.38-2\sqrt{4957}}{2.3362}$$

Divide 166.38-2\sqrt{4957} by 2.3362 by multiplying 166.38-2\sqrt{4957} by the reciprocal of 2.3362.

$$x=\frac{831900-10000\sqrt{4957}}{11681}$$

The equation is now solved.

$$x=\frac{10000\sqrt{4957}+831900}{11681} x=\frac{831900-10000\sqrt{4957}}{11681}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(100-x\right)^{2}.

$$10000-200x+x^{2}+\left(41+0.41x\right)^{2}=10000$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(41+0.41x\right)^{2}.

$$10000-200x+x^{2}+1681+33.62x+0.1681x^{2}=10000$$

Add 10000 and 1681 to get 11681.

$$11681-200x+x^{2}+33.62x+0.1681x^{2}=10000$$

Combine -200x and 33.62x to get -166.38x.

$$11681-166.38x+x^{2}+0.1681x^{2}=10000$$

Combine x^{2} and 0.1681x^{2} to get 1.1681x^{2}.

$$11681-166.38x+1.1681x^{2}=10000$$

Subtract 11681 from both sides.

$$-166.38x+1.1681x^{2}=10000-11681$$

Subtract 11681 from 10000 to get -1681.

$$-166.38x+1.1681x^{2}=-1681$$

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.

$$1.1681x^{2}-166.38x=-1681$$

Divide both sides of the equation by 1.1681, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{1.1681x^{2}-166.38x}{1.1681}=-\frac{1681}{1.1681}$$

Dividing by 1.1681 undoes the multiplication by 1.1681.

$$x^{2}+\left(-\frac{166.38}{1.1681}\right)x=-\frac{1681}{1.1681}$$

Divide -166.38 by 1.1681 by multiplying -166.38 by the reciprocal of 1.1681.

$$x^{2}-\frac{1663800}{11681}x=-\frac{1681}{1.1681}$$

Divide -1681 by 1.1681 by multiplying -1681 by the reciprocal of 1.1681.

$$x^{2}-\frac{1663800}{11681}x=-\frac{16810000}{11681}$$

Divide -\frac{1663800}{11681}, the coefficient of the x term, by 2 to get -\frac{831900}{11681}. Then add the square of -\frac{831900}{11681} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1663800}{11681}x+\left(-\frac{831900}{11681}\right)^{2}=-\frac{16810000}{11681}+\left(-\frac{831900}{11681}\right)^{2}$$

Square -\frac{831900}{11681} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1663800}{11681}x+\frac{692057610000}{136445761}=-\frac{16810000}{11681}+\frac{692057610000}{136445761}$$

Add -\frac{16810000}{11681} to \frac{692057610000}{136445761} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1663800}{11681}x+\frac{692057610000}{136445761}=\frac{495700000000}{136445761}$$

Factor x^{2}-\frac{1663800}{11681}x+\frac{692057610000}{136445761}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{831900}{11681}\right)^{2}=\frac{495700000000}{136445761}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{831900}{11681}\right)^{2}}=\sqrt{\frac{495700000000}{136445761}}$$

Simplify.

$$x-\frac{831900}{11681}=\frac{10000\sqrt{4957}}{11681} x-\frac{831900}{11681}=-\frac{10000\sqrt{4957}}{11681}$$

Add \frac{831900}{11681} to both sides of the equation.

$$x=\frac{10000\sqrt{4957}+831900}{11681} x=\frac{831900-10000\sqrt{4957}}{11681}$$