Solve sqrt{left(x-4right)^2+3^2}=0.8(9-x

$$\sqrt{ { \left(x-4 \right) }^{ 2 } + { 3 }^{ 2 } } =0.8(9-x$$

Solve for x

$$x = \frac{5 \sqrt{319} - 44}{9} \approx 5.033650611$$

$$x=\frac{-5\sqrt{319}-44}{9}\approx -14.811428389$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{\left(x-4\right)^{2}+3^{2}}\right)^{2}=\left(0.8\left(9-x\right)\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$\left(\sqrt{x^{2}-8x+16+3^{2}}\right)^{2}=\left(0.8\left(9-x\right)\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$\left(\sqrt{x^{2}-8x+16+9}\right)^{2}=\left(0.8\left(9-x\right)\right)^{2}$$

Add 16 and 9 to get 25.

$$\left(\sqrt{x^{2}-8x+25}\right)^{2}=\left(0.8\left(9-x\right)\right)^{2}$$

Calculate \sqrt{x^{2}-8x+25} to the power of 2 and get x^{2}-8x+25.

$$x^{2}-8x+25=\left(0.8\left(9-x\right)\right)^{2}$$

Use the distributive property to multiply 0.8 by 9-x.

$$x^{2}-8x+25=\left(7.2-0.8x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7.2-0.8x\right)^{2}.

$$x^{2}-8x+25=51.84-11.52x+0.64x^{2}$$

Subtract 51.84 from both sides.

$$x^{2}-8x+25-51.84=-11.52x+0.64x^{2}$$

Subtract 51.84 from 25 to get -26.84.

$$x^{2}-8x-26.84=-11.52x+0.64x^{2}$$

Add 11.52x to both sides.

$$x^{2}-8x-26.84+11.52x=0.64x^{2}$$

Combine -8x and 11.52x to get 3.52x.

$$x^{2}+3.52x-26.84=0.64x^{2}$$

Subtract 0.64x^{2} from both sides.

$$x^{2}+3.52x-26.84-0.64x^{2}=0$$

Combine x^{2} and -0.64x^{2} to get 0.36x^{2}.

$$0.36x^{2}+3.52x-26.84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.36 for a, 3.52 for b, and -26.84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3.52±\sqrt{3.52^{2}-4\times 0.36\left(-26.84\right)}}{2\times 0.36}$$

Square 3.52 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-3.52±\sqrt{12.3904-4\times 0.36\left(-26.84\right)}}{2\times 0.36}$$

Multiply -4 times 0.36.

$$x=\frac{-3.52±\sqrt{12.3904-1.44\left(-26.84\right)}}{2\times 0.36}$$

Multiply -1.44 times -26.84 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-3.52±\sqrt{\frac{7744+24156}{625}}}{2\times 0.36}$$

Add 12.3904 to 38.6496 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-3.52±\sqrt{51.04}}{2\times 0.36}$$

Take the square root of 51.04.

$$x=\frac{-3.52±\frac{2\sqrt{319}}{5}}{2\times 0.36}$$

Multiply 2 times 0.36.

$$x=\frac{-3.52±\frac{2\sqrt{319}}{5}}{0.72}$$

Now solve the equation x=\frac{-3.52±\frac{2\sqrt{319}}{5}}{0.72} when ± is plus. Add -3.52 to \frac{2\sqrt{319}}{5}.

$$x=\frac{\frac{2\sqrt{319}}{5}-\frac{88}{25}}{0.72}$$

Divide -\frac{88}{25}+\frac{2\sqrt{319}}{5} by 0.72 by multiplying -\frac{88}{25}+\frac{2\sqrt{319}}{5} by the reciprocal of 0.72.

$$x=\frac{5\sqrt{319}-44}{9}$$

Now solve the equation x=\frac{-3.52±\frac{2\sqrt{319}}{5}}{0.72} when ± is minus. Subtract \frac{2\sqrt{319}}{5} from -3.52.

$$x=\frac{-\frac{2\sqrt{319}}{5}-\frac{88}{25}}{0.72}$$

Divide -\frac{88}{25}-\frac{2\sqrt{319}}{5} by 0.72 by multiplying -\frac{88}{25}-\frac{2\sqrt{319}}{5} by the reciprocal of 0.72.

$$x=\frac{-5\sqrt{319}-44}{9}$$

The equation is now solved.

$$x=\frac{5\sqrt{319}-44}{9} x=\frac{-5\sqrt{319}-44}{9}$$

Substitute \frac{5\sqrt{319}-44}{9} for x in the equation \sqrt{\left(x-4\right)^{2}+3^{2}}=0.8\left(9-x\right).

$$\sqrt{\left(\frac{5\sqrt{319}-44}{9}-4\right)^{2}+3^{2}}=0.8\left(9-\frac{5\sqrt{319}-44}{9}\right)$$

Simplify. The value x=\frac{5\sqrt{319}-44}{9} satisfies the equation.

$$\frac{100}{9}-\frac{4}{9}\times 319^{\frac{1}{2}}=\frac{100}{9}-\frac{4}{9}\times 319^{\frac{1}{2}}$$

Substitute \frac{-5\sqrt{319}-44}{9} for x in the equation \sqrt{\left(x-4\right)^{2}+3^{2}}=0.8\left(9-x\right).

$$\sqrt{\left(\frac{-5\sqrt{319}-44}{9}-4\right)^{2}+3^{2}}=0.8\left(9-\frac{-5\sqrt{319}-44}{9}\right)$$

Simplify. The value x=\frac{-5\sqrt{319}-44}{9} satisfies the equation.

$$\frac{100}{9}+\frac{4}{9}\times 319^{\frac{1}{2}}=\frac{100}{9}+\frac{4}{9}\times 319^{\frac{1}{2}}$$

List all solutions of \sqrt{\left(x-4\right)^{2}+9}=\frac{4\left(9-x\right)}{5}.

$$x=\frac{5\sqrt{319}-44}{9} x=\frac{-5\sqrt{319}-44}{9}$$