Solve 0.84=(-x+sqrt{x^2+4*0.1*x})div2*0.1

$$0.84=(-x+ \sqrt{ { x }^{ 2 } +4 \times 0.1 \times x } ) \div 2 \times 0.1$$

Solve for x

$$x = -\frac{3528}{415} = -8\frac{208}{415} \approx -8.501204819$$

Solution Steps

Divide both sides by 0.1.

$$\frac{0.84}{0.1}=\frac{-x+\sqrt{x^{2}+4\times 0.1x}}{2}$$

Expand \frac{0.84}{0.1} by multiplying both numerator and the denominator by 100.

$$\frac{84}{10}=\frac{-x+\sqrt{x^{2}+4\times 0.1x}}{2}$$

Reduce the fraction \frac{84}{10} to lowest terms by extracting and canceling out 2.

$$\frac{42}{5}=\frac{-x+\sqrt{x^{2}+4\times 0.1x}}{2}$$

Multiply both sides by 2.

$$\frac{42}{5}\times 2=-x+\sqrt{x^{2}+4\times 0.1x}$$

Multiply \frac{42}{5} and 2 to get \frac{84}{5}.

$$\frac{84}{5}=-x+\sqrt{x^{2}+4\times 0.1x}$$

Multiply 4 and 0.1 to get 0.4.

$$\frac{84}{5}=-x+\sqrt{x^{2}+0.4x}$$

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

$$-x+\sqrt{x^{2}+0.4x}=\frac{84}{5}$$

Subtract -x from both sides of the equation.

$$\sqrt{x^{2}+0.4x}=\frac{84}{5}-\left(-x\right)$$

Multiply -1 and -1 to get 1.

$$\sqrt{x^{2}+0.4x}=\frac{84}{5}+x$$

Square both sides of the equation.

$$\left(\sqrt{x^{2}+0.4x}\right)^{2}=\left(\frac{84}{5}+x\right)^{2}$$

Calculate \sqrt{x^{2}+0.4x} to the power of 2 and get x^{2}+0.4x.

$$x^{2}+0.4x=\left(\frac{84}{5}+x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{84}{5}+x\right)^{2}.

$$x^{2}+0.4x=\frac{7056}{25}+\frac{168}{5}x+x^{2}$$

Subtract \frac{168}{5}x from both sides.

$$x^{2}+0.4x-\frac{168}{5}x=\frac{7056}{25}+x^{2}$$

Combine 0.4x and -\frac{168}{5}x to get -\frac{166}{5}x.

$$x^{2}-\frac{166}{5}x=\frac{7056}{25}+x^{2}$$

Subtract x^{2} from both sides.

$$x^{2}-\frac{166}{5}x-x^{2}=\frac{7056}{25}$$

Combine x^{2} and -x^{2} to get 0.

$$-\frac{166}{5}x=\frac{7056}{25}$$

Multiply both sides by -\frac{5}{166}, the reciprocal of -\frac{166}{5}.

$$x=\frac{7056}{25}\left(-\frac{5}{166}\right)$$

Multiply \frac{7056}{25} times -\frac{5}{166} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{7056\left(-5\right)}{25\times 166}$$

Do the multiplications in the fraction \frac{7056\left(-5\right)}{25\times 166}.

$$x=\frac{-35280}{4150}$$

Reduce the fraction \frac{-35280}{4150} to lowest terms by extracting and canceling out 10.

$$x=-\frac{3528}{415}$$

Substitute -\frac{3528}{415} for x in the equation 0.84=0.1\times \frac{-x+\sqrt{x^{2}+4\times 0.1x}}{2}.

$$0.84=0.1\times \frac{-\left(-\frac{3528}{415}\right)+\sqrt{\left(-\frac{3528}{415}\right)^{2}+4\times 0.1\left(-\frac{3528}{415}\right)}}{2}$$

Simplify. The value x=-\frac{3528}{415} satisfies the equation.

$$0.84=\frac{21}{25}$$

Equation \sqrt{x^{2}+\frac{2x}{5}}=x+\frac{84}{5} has a unique solution.

$$x=-\frac{3528}{415}$$