Solve 219.4^2=left(0.75*x+0.243*x*4right)^2+left(0.661*x+1.985*40right)^2

$${ 219.4 }^{ 2 } = { \left(0.75 \times x+0.243 \times x \times 4 \right) }^{ 2 } + { \left(0.661 \times x+1.985 \times 40 \right) }^{ 2 }$$

Solve for x

$$x = \frac{40 \sqrt{3626888670889} - 10496680}{680441} \approx 96.526940684$$

$$x=\frac{-40\sqrt{3626888670889}-10496680}{680441}\approx -127.379520114$$

Steps Using the Quadratic Formula

Calculate 219.4 to the power of 2 and get 48136.36.

$$48136.36=\left(0.75x+0.243x\times 4\right)^{2}+\left(0.661x+1.985\times 40\right)^{2}$$

Multiply 0.243 and 4 to get 0.972.

$$48136.36=\left(0.75x+0.972x\right)^{2}+\left(0.661x+1.985\times 40\right)^{2}$$

Combine 0.75x and 0.972x to get 1.722x.

$$48136.36=\left(1.722x\right)^{2}+\left(0.661x+1.985\times 40\right)^{2}$$

Expand \left(1.722x\right)^{2}.

$$48136.36=1.722^{2}x^{2}+\left(0.661x+1.985\times 40\right)^{2}$$

Calculate 1.722 to the power of 2 and get 2.965284.

$$48136.36=2.965284x^{2}+\left(0.661x+1.985\times 40\right)^{2}$$

Multiply 1.985 and 40 to get 79.4.

$$48136.36=2.965284x^{2}+\left(0.661x+79.4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(0.661x+79.4\right)^{2}.

$$48136.36=2.965284x^{2}+0.436921x^{2}+104.9668x+6304.36$$

Combine 2.965284x^{2} and 0.436921x^{2} to get 3.402205x^{2}.

$$48136.36=3.402205x^{2}+104.9668x+6304.36$$

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

$$3.402205x^{2}+104.9668x+6304.36=48136.36$$

Subtract 48136.36 from both sides.

$$3.402205x^{2}+104.9668x+6304.36-48136.36=0$$

Subtract 48136.36 from 6304.36 to get -41832.

$$3.402205x^{2}+104.9668x-41832=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3.402205 for a, 104.9668 for b, and -41832 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-104.9668±\sqrt{104.9668^{2}-4\times 3.402205\left(-41832\right)}}{2\times 3.402205}$$

Square 104.9668 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-104.9668±\sqrt{11018.02910224-4\times 3.402205\left(-41832\right)}}{2\times 3.402205}$$

Multiply -4 times 3.402205.

$$x=\frac{-104.9668±\sqrt{11018.02910224-13.60882\left(-41832\right)}}{2\times 3.402205}$$

Multiply -13.60882 times -41832.

$$x=\frac{-104.9668±\sqrt{11018.02910224+569284.15824}}{2\times 3.402205}$$

Add 11018.02910224 to 569284.15824 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-104.9668±\sqrt{580302.18734224}}{2\times 3.402205}$$

Take the square root of 580302.18734224.

$$x=\frac{-104.9668±\frac{\sqrt{3626888670889}}{2500}}{2\times 3.402205}$$

Multiply 2 times 3.402205.

$$x=\frac{-104.9668±\frac{\sqrt{3626888670889}}{2500}}{6.80441}$$

Now solve the equation x=\frac{-104.9668±\frac{\sqrt{3626888670889}}{2500}}{6.80441} when ± is plus. Add -104.9668 to \frac{\sqrt{3626888670889}}{2500}.

$$x=\frac{\sqrt{3626888670889}-262417}{6.80441\times 2500}$$

Divide \frac{-262417+\sqrt{3626888670889}}{2500} by 6.80441 by multiplying \frac{-262417+\sqrt{3626888670889}}{2500} by the reciprocal of 6.80441.

$$x=\frac{40\sqrt{3626888670889}-10496680}{680441}$$

Now solve the equation x=\frac{-104.9668±\frac{\sqrt{3626888670889}}{2500}}{6.80441} when ± is minus. Subtract \frac{\sqrt{3626888670889}}{2500} from -104.9668.

$$x=\frac{-\sqrt{3626888670889}-262417}{6.80441\times 2500}$$

Divide \frac{-262417-\sqrt{3626888670889}}{2500} by 6.80441 by multiplying \frac{-262417-\sqrt{3626888670889}}{2500} by the reciprocal of 6.80441.

$$x=\frac{-40\sqrt{3626888670889}-10496680}{680441}$$

The equation is now solved.

$$x=\frac{40\sqrt{3626888670889}-10496680}{680441} x=\frac{-40\sqrt{3626888670889}-10496680}{680441}$$

Steps for Completing the Square

Calculate 219.4 to the power of 2 and get 48136.36.

$$48136.36=\left(0.75x+0.243x\times 4\right)^{2}+\left(0.661x+1.985\times 40\right)^{2}$$

Multiply 0.243 and 4 to get 0.972.

$$48136.36=\left(0.75x+0.972x\right)^{2}+\left(0.661x+1.985\times 40\right)^{2}$$

Combine 0.75x and 0.972x to get 1.722x.

$$48136.36=\left(1.722x\right)^{2}+\left(0.661x+1.985\times 40\right)^{2}$$

Expand \left(1.722x\right)^{2}.

$$48136.36=1.722^{2}x^{2}+\left(0.661x+1.985\times 40\right)^{2}$$

Calculate 1.722 to the power of 2 and get 2.965284.

$$48136.36=2.965284x^{2}+\left(0.661x+1.985\times 40\right)^{2}$$

Multiply 1.985 and 40 to get 79.4.

$$48136.36=2.965284x^{2}+\left(0.661x+79.4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(0.661x+79.4\right)^{2}.

$$48136.36=2.965284x^{2}+0.436921x^{2}+104.9668x+6304.36$$

Combine 2.965284x^{2} and 0.436921x^{2} to get 3.402205x^{2}.

$$48136.36=3.402205x^{2}+104.9668x+6304.36$$

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

$$3.402205x^{2}+104.9668x+6304.36=48136.36$$

Subtract 6304.36 from both sides.

$$3.402205x^{2}+104.9668x=48136.36-6304.36$$

Subtract 6304.36 from 48136.36 to get 41832.

$$3.402205x^{2}+104.9668x=41832$$

Divide both sides of the equation by 3.402205, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{3.402205x^{2}+104.9668x}{3.402205}=\frac{41832}{3.402205}$$

Dividing by 3.402205 undoes the multiplication by 3.402205.

$$x^{2}+\frac{104.9668}{3.402205}x=\frac{41832}{3.402205}$$

Divide 104.9668 by 3.402205 by multiplying 104.9668 by the reciprocal of 3.402205.

$$x^{2}+\frac{20993360}{680441}x=\frac{41832}{3.402205}$$

Divide 41832 by 3.402205 by multiplying 41832 by the reciprocal of 3.402205.

$$x^{2}+\frac{20993360}{680441}x=\frac{8366400000}{680441}$$

Divide \frac{20993360}{680441}, the coefficient of the x term, by 2 to get \frac{10496680}{680441}. Then add the square of \frac{10496680}{680441} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{20993360}{680441}x+\frac{10496680}{680441}^{2}=\frac{8366400000}{680441}+\frac{10496680}{680441}^{2}$$

Square \frac{10496680}{680441} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{20993360}{680441}x+\frac{110180291022400}{462999954481}=\frac{8366400000}{680441}+\frac{110180291022400}{462999954481}$$

Add \frac{8366400000}{680441} to \frac{110180291022400}{462999954481} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{20993360}{680441}x+\frac{110180291022400}{462999954481}=\frac{5803021873422400}{462999954481}$$

Factor x^{2}+\frac{20993360}{680441}x+\frac{110180291022400}{462999954481}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{10496680}{680441}\right)^{2}=\frac{5803021873422400}{462999954481}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{10496680}{680441}\right)^{2}}=\sqrt{\frac{5803021873422400}{462999954481}}$$

Simplify.

$$x+\frac{10496680}{680441}=\frac{40\sqrt{3626888670889}}{680441} x+\frac{10496680}{680441}=-\frac{40\sqrt{3626888670889}}{680441}$$

Subtract \frac{10496680}{680441} from both sides of the equation.

$$x=\frac{40\sqrt{3626888670889}-10496680}{680441} x=\frac{-40\sqrt{3626888670889}-10496680}{680441}$$