Solve 3.9quadsqrt{4^2+x^2}=8-x

$$3.9 \quad \sqrt { 4 ^ { 2 } + x ^ { 2 } } = 8 - x$$

Solve for x (complex solution)

$$x=\frac{-i\times 156\sqrt{1021}-800}{1421}\approx -0.562983814-3.507869202i$$

Solution Steps

Square both sides of the equation.

$$\left(3.9\sqrt{4^{2}+x^{2}}\right)^{2}=\left(8-x\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$\left(3.9\sqrt{16+x^{2}}\right)^{2}=\left(8-x\right)^{2}$$

Expand \left(3.9\sqrt{16+x^{2}}\right)^{2}.

$$3.9^{2}\left(\sqrt{16+x^{2}}\right)^{2}=\left(8-x\right)^{2}$$

Calculate 3.9 to the power of 2 and get 15.21.

$$15.21\left(\sqrt{16+x^{2}}\right)^{2}=\left(8-x\right)^{2}$$

Calculate \sqrt{16+x^{2}} to the power of 2 and get 16+x^{2}.

$$15.21\left(16+x^{2}\right)=\left(8-x\right)^{2}$$

Use the distributive property to multiply 15.21 by 16+x^{2}.

$$243.36+15.21x^{2}=\left(8-x\right)^{2}$$

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

$$243.36+15.21x^{2}=64-16x+x^{2}$$

Subtract 64 from both sides.

$$243.36+15.21x^{2}-64=-16x+x^{2}$$

Subtract 64 from 243.36 to get 179.36.

$$179.36+15.21x^{2}=-16x+x^{2}$$

Add 16x to both sides.

$$179.36+15.21x^{2}+16x=x^{2}$$

Subtract x^{2} from both sides.

$$179.36+15.21x^{2}+16x-x^{2}=0$$

Combine 15.21x^{2} and -x^{2} to get 14.21x^{2}.

$$179.36+14.21x^{2}+16x=0$$

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.

$$14.21x^{2}+16x+179.36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 14.21 for a, 16 for b, and 179.36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 14.21\times 179.36}}{2\times 14.21}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 14.21\times 179.36}}{2\times 14.21}$$

Multiply -4 times 14.21.

$$x=\frac{-16±\sqrt{256-56.84\times 179.36}}{2\times 14.21}$$

Multiply -56.84 times 179.36 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-16±\sqrt{256-10194.8224}}{2\times 14.21}$$

Add 256 to -10194.8224.

$$x=\frac{-16±\sqrt{-9938.8224}}{2\times 14.21}$$

Take the square root of -9938.8224.

$$x=\frac{-16±\frac{78\sqrt{1021}i}{25}}{2\times 14.21}$$

Multiply 2 times 14.21.

$$x=\frac{-16±\frac{78\sqrt{1021}i}{25}}{28.42}$$

Now solve the equation x=\frac{-16±\frac{78\sqrt{1021}i}{25}}{28.42} when ± is plus. Add -16 to \frac{78i\sqrt{1021}}{25}.

$$x=\frac{\frac{78\sqrt{1021}i}{25}-16}{28.42}$$

Divide -16+\frac{78i\sqrt{1021}}{25} by 28.42 by multiplying -16+\frac{78i\sqrt{1021}}{25} by the reciprocal of 28.42.

$$x=\frac{-800+156\sqrt{1021}i}{1421}$$

Now solve the equation x=\frac{-16±\frac{78\sqrt{1021}i}{25}}{28.42} when ± is minus. Subtract \frac{78i\sqrt{1021}}{25} from -16.

$$x=\frac{-\frac{78\sqrt{1021}i}{25}-16}{28.42}$$

Divide -16-\frac{78i\sqrt{1021}}{25} by 28.42 by multiplying -16-\frac{78i\sqrt{1021}}{25} by the reciprocal of 28.42.

$$x=\frac{-156\sqrt{1021}i-800}{1421}$$

The equation is now solved.

$$x=\frac{-800+156\sqrt{1021}i}{1421} x=\frac{-156\sqrt{1021}i-800}{1421}$$

Substitute \frac{-800+156\sqrt{1021}i}{1421} for x in the equation 3.9\sqrt{4^{2}+x^{2}}=8-x.

$$3.9\sqrt{4^{2}+\left(\frac{-800+156\sqrt{1021}i}{1421}\right)^{2}}=8-\frac{-800+156\sqrt{1021}i}{1421}$$

Simplify. The value x=\frac{-800+156\sqrt{1021}i}{1421} does not satisfy the equation.

$$-\frac{12168}{1421}+\frac{156}{1421}i\times 1021^{\frac{1}{2}}=\frac{12168}{1421}-\frac{156}{1421}i\times 1021^{\frac{1}{2}}$$

Substitute \frac{-156\sqrt{1021}i-800}{1421} for x in the equation 3.9\sqrt{4^{2}+x^{2}}=8-x.

$$3.9\sqrt{4^{2}+\left(\frac{-156\sqrt{1021}i-800}{1421}\right)^{2}}=8-\frac{-156\sqrt{1021}i-800}{1421}$$

Simplify. The value x=\frac{-156\sqrt{1021}i-800}{1421} satisfies the equation.

$$\frac{12168}{1421}+\frac{156}{1421}i\times 1021^{\frac{1}{2}}=\frac{12168}{1421}+\frac{156}{1421}i\times 1021^{\frac{1}{2}}$$

Equation \frac{39\sqrt{x^{2}+16}}{10}=8-x has a unique solution.

$$x=\frac{-156\sqrt{1021}i-800}{1421}$$