Solve 200=1.1x+.06x^2

$$200 = 1.1 x + .06 x ^ { 2 }$$

Solve for x

$$x = \frac{5 \sqrt{4921} - 55}{6} \approx 49.291533024$$

$$x=\frac{-5\sqrt{4921}-55}{6}\approx -67.624866358$$

Steps Using the Quadratic Formula

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

$$1.1x+0.06x^{2}=200$$

Subtract 200 from both sides.

$$1.1x+0.06x^{2}-200=0$$

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.

$$0.06x^{2}+1.1x-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.06 for a, 1.1 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1.1±\sqrt{1.1^{2}-4\times 0.06\left(-200\right)}}{2\times 0.06}$$

Square 1.1 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-1.1±\sqrt{1.21-4\times 0.06\left(-200\right)}}{2\times 0.06}$$

Multiply -4 times 0.06.

$$x=\frac{-1.1±\sqrt{1.21-0.24\left(-200\right)}}{2\times 0.06}$$

Multiply -0.24 times -200.

$$x=\frac{-1.1±\sqrt{1.21+48}}{2\times 0.06}$$

Add 1.21 to 48.

$$x=\frac{-1.1±\sqrt{49.21}}{2\times 0.06}$$

Take the square root of 49.21.

$$x=\frac{-1.1±\frac{\sqrt{4921}}{10}}{2\times 0.06}$$

Multiply 2 times 0.06.

$$x=\frac{-1.1±\frac{\sqrt{4921}}{10}}{0.12}$$

Now solve the equation x=\frac{-1.1±\frac{\sqrt{4921}}{10}}{0.12} when ± is plus. Add -1.1 to \frac{\sqrt{4921}}{10}.

$$x=\frac{\sqrt{4921}-11}{0.12\times 10}$$

Divide \frac{-11+\sqrt{4921}}{10} by 0.12 by multiplying \frac{-11+\sqrt{4921}}{10} by the reciprocal of 0.12.

$$x=\frac{5\sqrt{4921}-55}{6}$$

Now solve the equation x=\frac{-1.1±\frac{\sqrt{4921}}{10}}{0.12} when ± is minus. Subtract \frac{\sqrt{4921}}{10} from -1.1.

$$x=\frac{-\sqrt{4921}-11}{0.12\times 10}$$

Divide \frac{-11-\sqrt{4921}}{10} by 0.12 by multiplying \frac{-11-\sqrt{4921}}{10} by the reciprocal of 0.12.

$$x=\frac{-5\sqrt{4921}-55}{6}$$

The equation is now solved.

$$x=\frac{5\sqrt{4921}-55}{6} x=\frac{-5\sqrt{4921}-55}{6}$$

Steps for Completing the Square

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

$$1.1x+0.06x^{2}=200$$

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.

$$0.06x^{2}+1.1x=200$$

Divide both sides of the equation by 0.06, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.06x^{2}+1.1x}{0.06}=\frac{200}{0.06}$$

Dividing by 0.06 undoes the multiplication by 0.06.

$$x^{2}+\frac{1.1}{0.06}x=\frac{200}{0.06}$$

Divide 1.1 by 0.06 by multiplying 1.1 by the reciprocal of 0.06.

$$x^{2}+\frac{55}{3}x=\frac{200}{0.06}$$

Divide 200 by 0.06 by multiplying 200 by the reciprocal of 0.06.

$$x^{2}+\frac{55}{3}x=\frac{10000}{3}$$

Divide \frac{55}{3}, the coefficient of the x term, by 2 to get \frac{55}{6}. Then add the square of \frac{55}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{55}{3}x+\frac{55}{6}^{2}=\frac{10000}{3}+\frac{55}{6}^{2}$$

Square \frac{55}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{55}{3}x+\frac{3025}{36}=\frac{10000}{3}+\frac{3025}{36}$$

Add \frac{10000}{3} to \frac{3025}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{55}{3}x+\frac{3025}{36}=\frac{123025}{36}$$

Factor x^{2}+\frac{55}{3}x+\frac{3025}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{55}{6}\right)^{2}=\frac{123025}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{55}{6}\right)^{2}}=\sqrt{\frac{123025}{36}}$$

Simplify.

$$x+\frac{55}{6}=\frac{5\sqrt{4921}}{6} x+\frac{55}{6}=-\frac{5\sqrt{4921}}{6}$$

Subtract \frac{55}{6} from both sides of the equation.

$$x=\frac{5\sqrt{4921}-55}{6} x=\frac{-5\sqrt{4921}-55}{6}$$