Solve 0.5x=1-0.6x+0.09x^2

$$0.5x = 1-0.6x+0.09 { x }^{ 2 }$$

Solve for x

$$x = \frac{5 \sqrt{85} + 55}{9} \approx 11.233080254$$

$$x=\frac{55-5\sqrt{85}}{9}\approx 0.989141968$$

Steps Using the Quadratic Formula

Subtract 1 from both sides.

$$0.5x-1=-0.6x+0.09x^{2}$$

Add 0.6x to both sides.

$$0.5x-1+0.6x=0.09x^{2}$$

Combine 0.5x and 0.6x to get 1.1x.

$$1.1x-1=0.09x^{2}$$

Subtract 0.09x^{2} from both sides.

$$1.1x-1-0.09x^{2}=0$$

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.

$$-0.09x^{2}+1.1x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.09 for a, 1.1 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1.1±\sqrt{1.1^{2}-4\left(-0.09\right)\left(-1\right)}}{2\left(-0.09\right)}$$

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

$$x=\frac{-1.1±\sqrt{1.21-4\left(-0.09\right)\left(-1\right)}}{2\left(-0.09\right)}$$

Multiply -4 times -0.09.

$$x=\frac{-1.1±\sqrt{1.21+0.36\left(-1\right)}}{2\left(-0.09\right)}$$

Multiply 0.36 times -1.

$$x=\frac{-1.1±\sqrt{1.21-0.36}}{2\left(-0.09\right)}$$

Add 1.21 to -0.36 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1.1±\sqrt{0.85}}{2\left(-0.09\right)}$$

Take the square root of 0.85.

$$x=\frac{-1.1±\frac{\sqrt{85}}{10}}{2\left(-0.09\right)}$$

Multiply 2 times -0.09.

$$x=\frac{-1.1±\frac{\sqrt{85}}{10}}{-0.18}$$

Now solve the equation x=\frac{-1.1±\frac{\sqrt{85}}{10}}{-0.18} when ± is plus. Add -1.1 to \frac{\sqrt{85}}{10}.

$$x=\frac{\sqrt{85}-11}{-0.18\times 10}$$

Divide \frac{-11+\sqrt{85}}{10} by -0.18 by multiplying \frac{-11+\sqrt{85}}{10} by the reciprocal of -0.18.

$$x=\frac{55-5\sqrt{85}}{9}$$

Now solve the equation x=\frac{-1.1±\frac{\sqrt{85}}{10}}{-0.18} when ± is minus. Subtract \frac{\sqrt{85}}{10} from -1.1.

$$x=\frac{-\sqrt{85}-11}{-0.18\times 10}$$

Divide \frac{-11-\sqrt{85}}{10} by -0.18 by multiplying \frac{-11-\sqrt{85}}{10} by the reciprocal of -0.18.

$$x=\frac{5\sqrt{85}+55}{9}$$

The equation is now solved.

$$x=\frac{55-5\sqrt{85}}{9} x=\frac{5\sqrt{85}+55}{9}$$

Steps for Completing the Square

Add 0.6x to both sides.

$$0.5x+0.6x=1+0.09x^{2}$$

Combine 0.5x and 0.6x to get 1.1x.

$$1.1x=1+0.09x^{2}$$

Subtract 0.09x^{2} from both sides.

$$1.1x-0.09x^{2}=1$$

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.

$$-0.09x^{2}+1.1x=1$$

Divide both sides of the equation by -0.09, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.09x^{2}+1.1x}{-0.09}=\frac{1}{-0.09}$$

Dividing by -0.09 undoes the multiplication by -0.09.

$$x^{2}+\frac{1.1}{-0.09}x=\frac{1}{-0.09}$$

Divide 1.1 by -0.09 by multiplying 1.1 by the reciprocal of -0.09.

$$x^{2}-\frac{110}{9}x=\frac{1}{-0.09}$$

Divide 1 by -0.09 by multiplying 1 by the reciprocal of -0.09.

$$x^{2}-\frac{110}{9}x=-\frac{100}{9}$$

Divide -\frac{110}{9}, the coefficient of the x term, by 2 to get -\frac{55}{9}. Then add the square of -\frac{55}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{110}{9}x+\left(-\frac{55}{9}\right)^{2}=-\frac{100}{9}+\left(-\frac{55}{9}\right)^{2}$$

Square -\frac{55}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{110}{9}x+\frac{3025}{81}=-\frac{100}{9}+\frac{3025}{81}$$

Add -\frac{100}{9} to \frac{3025}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{110}{9}x+\frac{3025}{81}=\frac{2125}{81}$$

Factor x^{2}-\frac{110}{9}x+\frac{3025}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{55}{9}\right)^{2}=\frac{2125}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{55}{9}\right)^{2}}=\sqrt{\frac{2125}{81}}$$

Simplify.

$$x-\frac{55}{9}=\frac{5\sqrt{85}}{9} x-\frac{55}{9}=-\frac{5\sqrt{85}}{9}$$

Add \frac{55}{9} to both sides of the equation.

$$x=\frac{5\sqrt{85}+55}{9} x=\frac{55-5\sqrt{85}}{9}$$