Solve 0.011x^2-0.33x-109=0

$$0.011 { x }^{ 2 } -0.33x-109=0$$

Solve for x

$$x=\frac{35\sqrt{1001}}{11}+15\approx 115.668221943$$

$$x=-\frac{35\sqrt{1001}}{11}+15\approx -85.668221943$$

Steps Using the Quadratic Formula

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.

$$0.011x^{2}-0.33x-109=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.011 for a, -0.33 for b, and -109 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-0.33\right)±\sqrt{\left(-0.33\right)^{2}-4\times 0.011\left(-109\right)}}{2\times 0.011}$$

Square -0.33 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-0.33\right)±\sqrt{0.1089-4\times 0.011\left(-109\right)}}{2\times 0.011}$$

Multiply -4 times 0.011.

$$x=\frac{-\left(-0.33\right)±\sqrt{0.1089-0.044\left(-109\right)}}{2\times 0.011}$$

Multiply -0.044 times -109.

$$x=\frac{-\left(-0.33\right)±\sqrt{0.1089+4.796}}{2\times 0.011}$$

Add 0.1089 to 4.796 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.33\right)±\sqrt{4.9049}}{2\times 0.011}$$

Take the square root of 4.9049.

$$x=\frac{-\left(-0.33\right)±\frac{7\sqrt{1001}}{100}}{2\times 0.011}$$

The opposite of -0.33 is 0.33.

$$x=\frac{0.33±\frac{7\sqrt{1001}}{100}}{2\times 0.011}$$

Multiply 2 times 0.011.

$$x=\frac{0.33±\frac{7\sqrt{1001}}{100}}{0.022}$$

Now solve the equation x=\frac{0.33±\frac{7\sqrt{1001}}{100}}{0.022} when ± is plus. Add 0.33 to \frac{7\sqrt{1001}}{100}.

$$x=\frac{7\sqrt{1001}+33}{0.022\times 100}$$

Divide \frac{33+7\sqrt{1001}}{100} by 0.022 by multiplying \frac{33+7\sqrt{1001}}{100} by the reciprocal of 0.022.

$$x=\frac{35\sqrt{1001}}{11}+15$$

Now solve the equation x=\frac{0.33±\frac{7\sqrt{1001}}{100}}{0.022} when ± is minus. Subtract \frac{7\sqrt{1001}}{100} from 0.33.

$$x=\frac{33-7\sqrt{1001}}{0.022\times 100}$$

Divide \frac{33-7\sqrt{1001}}{100} by 0.022 by multiplying \frac{33-7\sqrt{1001}}{100} by the reciprocal of 0.022.

$$x=-\frac{35\sqrt{1001}}{11}+15$$

The equation is now solved.

$$x=\frac{35\sqrt{1001}}{11}+15 x=-\frac{35\sqrt{1001}}{11}+15$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$0.011x^{2}-0.33x-109=0$$

Add 109 to both sides of the equation.

$$0.011x^{2}-0.33x-109-\left(-109\right)=-\left(-109\right)$$

Subtracting -109 from itself leaves 0.

$$0.011x^{2}-0.33x=-\left(-109\right)$$

Subtract -109 from 0.

$$0.011x^{2}-0.33x=109$$

Divide both sides of the equation by 0.011, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.011x^{2}-0.33x}{0.011}=\frac{109}{0.011}$$

Dividing by 0.011 undoes the multiplication by 0.011.

$$x^{2}+\left(-\frac{0.33}{0.011}\right)x=\frac{109}{0.011}$$

Divide -0.33 by 0.011 by multiplying -0.33 by the reciprocal of 0.011.

$$x^{2}-30x=\frac{109}{0.011}$$

Divide 109 by 0.011 by multiplying 109 by the reciprocal of 0.011.

$$x^{2}-30x=\frac{109000}{11}$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-30x+\left(-15\right)^{2}=\frac{109000}{11}+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=\frac{109000}{11}+225$$

Add \frac{109000}{11} to 225.

$$x^{2}-30x+225=\frac{111475}{11}$$

Factor x^{2}-30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-15\right)^{2}=\frac{111475}{11}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{\frac{111475}{11}}$$

Simplify.

$$x-15=\frac{35\sqrt{1001}}{11} x-15=-\frac{35\sqrt{1001}}{11}$$

Add 15 to both sides of the equation.

$$x=\frac{35\sqrt{1001}}{11}+15 x=-\frac{35\sqrt{1001}}{11}+15$$