Solve 9%+1%*x^2*9+sqrt{4+4+6+2}+9x=0

$$9 \% +1 \% \times { x }^{ 2 } \times 9+ \sqrt{ 4+4+6+2 } +9x=0$$

Solve for x

$$x=\frac{\sqrt{22091}}{3}-50\approx -0.456528628$$

$$x=-\frac{\sqrt{22091}}{3}-50\approx -99.543471372$$

Steps Using the Quadratic Formula

Multiply \frac{1}{100} and 9 to get \frac{9}{100}.

$$\frac{9}{100}+\frac{9}{100}x^{2}+\sqrt{4+4+6+2}+9x=0$$

Add 4 and 4 to get 8.

$$\frac{9}{100}+\frac{9}{100}x^{2}+\sqrt{8+6+2}+9x=0$$

Add 8 and 6 to get 14.

$$\frac{9}{100}+\frac{9}{100}x^{2}+\sqrt{14+2}+9x=0$$

Add 14 and 2 to get 16.

$$\frac{9}{100}+\frac{9}{100}x^{2}+\sqrt{16}+9x=0$$

Calculate the square root of 16 and get 4.

$$\frac{9}{100}+\frac{9}{100}x^{2}+4+9x=0$$

Add \frac{9}{100} and 4 to get \frac{409}{100}.

$$\frac{409}{100}+\frac{9}{100}x^{2}+9x=0$$

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.

$$\frac{9}{100}x^{2}+9x+\frac{409}{100}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{9}{100} for a, 9 for b, and \frac{409}{100} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\times \frac{9}{100}\times \frac{409}{100}}}{2\times \frac{9}{100}}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\times \frac{9}{100}\times \frac{409}{100}}}{2\times \frac{9}{100}}$$

Multiply -4 times \frac{9}{100}.

$$x=\frac{-9±\sqrt{81-\frac{9}{25}\times \frac{409}{100}}}{2\times \frac{9}{100}}$$

Multiply -\frac{9}{25} times \frac{409}{100} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-9±\sqrt{81-\frac{3681}{2500}}}{2\times \frac{9}{100}}$$

Add 81 to -\frac{3681}{2500}.

$$x=\frac{-9±\sqrt{\frac{198819}{2500}}}{2\times \frac{9}{100}}$$

Take the square root of \frac{198819}{2500}.

$$x=\frac{-9±\frac{3\sqrt{22091}}{50}}{2\times \frac{9}{100}}$$

Multiply 2 times \frac{9}{100}.

$$x=\frac{-9±\frac{3\sqrt{22091}}{50}}{\frac{9}{50}}$$

Now solve the equation x=\frac{-9±\frac{3\sqrt{22091}}{50}}{\frac{9}{50}} when ± is plus. Add -9 to \frac{3\sqrt{22091}}{50}.

$$x=\frac{\frac{3\sqrt{22091}}{50}-9}{\frac{9}{50}}$$

Divide -9+\frac{3\sqrt{22091}}{50} by \frac{9}{50} by multiplying -9+\frac{3\sqrt{22091}}{50} by the reciprocal of \frac{9}{50}.

$$x=\frac{\sqrt{22091}}{3}-50$$

Now solve the equation x=\frac{-9±\frac{3\sqrt{22091}}{50}}{\frac{9}{50}} when ± is minus. Subtract \frac{3\sqrt{22091}}{50} from -9.

$$x=\frac{-\frac{3\sqrt{22091}}{50}-9}{\frac{9}{50}}$$

Divide -9-\frac{3\sqrt{22091}}{50} by \frac{9}{50} by multiplying -9-\frac{3\sqrt{22091}}{50} by the reciprocal of \frac{9}{50}.

$$x=-\frac{\sqrt{22091}}{3}-50$$

The equation is now solved.

$$x=\frac{\sqrt{22091}}{3}-50 x=-\frac{\sqrt{22091}}{3}-50$$

Steps for Completing the Square

Multiply \frac{1}{100} and 9 to get \frac{9}{100}.

$$\frac{9}{100}+\frac{9}{100}x^{2}+\sqrt{4+4+6+2}+9x=0$$

Add 4 and 4 to get 8.

$$\frac{9}{100}+\frac{9}{100}x^{2}+\sqrt{8+6+2}+9x=0$$

Add 8 and 6 to get 14.

$$\frac{9}{100}+\frac{9}{100}x^{2}+\sqrt{14+2}+9x=0$$

Add 14 and 2 to get 16.

$$\frac{9}{100}+\frac{9}{100}x^{2}+\sqrt{16}+9x=0$$

Calculate the square root of 16 and get 4.

$$\frac{9}{100}+\frac{9}{100}x^{2}+4+9x=0$$

Add \frac{9}{100} and 4 to get \frac{409}{100}.

$$\frac{409}{100}+\frac{9}{100}x^{2}+9x=0$$

Subtract \frac{409}{100} from both sides. Anything subtracted from zero gives its negation.

$$\frac{9}{100}x^{2}+9x=-\frac{409}{100}$$

Divide both sides of the equation by \frac{9}{100}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{9}{100}x^{2}+9x}{\frac{9}{100}}=-\frac{\frac{409}{100}}{\frac{9}{100}}$$

Dividing by \frac{9}{100} undoes the multiplication by \frac{9}{100}.

$$x^{2}+\frac{9}{\frac{9}{100}}x=-\frac{\frac{409}{100}}{\frac{9}{100}}$$

Divide 9 by \frac{9}{100} by multiplying 9 by the reciprocal of \frac{9}{100}.

$$x^{2}+100x=-\frac{\frac{409}{100}}{\frac{9}{100}}$$

Divide -\frac{409}{100} by \frac{9}{100} by multiplying -\frac{409}{100} by the reciprocal of \frac{9}{100}.

$$x^{2}+100x=-\frac{409}{9}$$

Divide 100, the coefficient of the x term, by 2 to get 50. Then add the square of 50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+100x+50^{2}=-\frac{409}{9}+50^{2}$$

Square 50.

$$x^{2}+100x+2500=-\frac{409}{9}+2500$$

Add -\frac{409}{9} to 2500.

$$x^{2}+100x+2500=\frac{22091}{9}$$

Factor x^{2}+100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+50\right)^{2}=\frac{22091}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+50\right)^{2}}=\sqrt{\frac{22091}{9}}$$

Simplify.

$$x+50=\frac{\sqrt{22091}}{3} x+50=-\frac{\sqrt{22091}}{3}$$

Subtract 50 from both sides of the equation.

$$x=\frac{\sqrt{22091}}{3}-50 x=-\frac{\sqrt{22091}}{3}-50$$