Solve 0.21x=0.1+0.09x^2

$$0.21 x = 0.1 + 0.09 x ^ { 2 }$$

Solve for x

$$x=\frac{2}{3}\approx 0.666666667$$

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Steps Using the Quadratic Formula

Subtract 0.1 from both sides.

$$0.21x-0.1=0.09x^{2}$$

Subtract 0.09x^{2} from both sides.

$$0.21x-0.1-0.09x^{2}=0$$

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.

$$-0.09x^{2}+0.21x-0.1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.09 for a, 0.21 for b, and -0.1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.21±\sqrt{0.21^{2}-4\left(-0.09\right)\left(-0.1\right)}}{2\left(-0.09\right)}$$

Square 0.21 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.21±\sqrt{0.0441-4\left(-0.09\right)\left(-0.1\right)}}{2\left(-0.09\right)}$$

Multiply -4 times -0.09.

$$x=\frac{-0.21±\sqrt{0.0441+0.36\left(-0.1\right)}}{2\left(-0.09\right)}$$

Multiply 0.36 times -0.1 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.21±\sqrt{0.0441-0.036}}{2\left(-0.09\right)}$$

Add 0.0441 to -0.036 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.21±\sqrt{0.0081}}{2\left(-0.09\right)}$$

Take the square root of 0.0081.

$$x=\frac{-0.21±\frac{9}{100}}{2\left(-0.09\right)}$$

Multiply 2 times -0.09.

$$x=\frac{-0.21±\frac{9}{100}}{-0.18}$$

Now solve the equation x=\frac{-0.21±\frac{9}{100}}{-0.18} when ± is plus. Add -0.21 to \frac{9}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{3}{25}}{-0.18}$$

Divide -\frac{3}{25} by -0.18 by multiplying -\frac{3}{25} by the reciprocal of -0.18.

$$x=\frac{2}{3}$$

Now solve the equation x=\frac{-0.21±\frac{9}{100}}{-0.18} when ± is minus. Subtract \frac{9}{100} from -0.21 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{3}{10}}{-0.18}$$

Divide -\frac{3}{10} by -0.18 by multiplying -\frac{3}{10} by the reciprocal of -0.18.

$$x=\frac{5}{3}$$

The equation is now solved.

$$x=\frac{2}{3} x=\frac{5}{3}$$

Steps for Completing the Square

Subtract 0.09x^{2} from both sides.

$$0.21x-0.09x^{2}=0.1$$

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.

$$-0.09x^{2}+0.21x=0.1$$

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

$$\frac{-0.09x^{2}+0.21x}{-0.09}=\frac{0.1}{-0.09}$$

Dividing by -0.09 undoes the multiplication by -0.09.

$$x^{2}+\frac{0.21}{-0.09}x=\frac{0.1}{-0.09}$$

Divide 0.21 by -0.09 by multiplying 0.21 by the reciprocal of -0.09.

$$x^{2}-\frac{7}{3}x=\frac{0.1}{-0.09}$$

Divide 0.1 by -0.09 by multiplying 0.1 by the reciprocal of -0.09.

$$x^{2}-\frac{7}{3}x=-\frac{10}{9}$$

Divide -\frac{7}{3}, the coefficient of the x term, by 2 to get -\frac{7}{6}. Then add the square of -\frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{3}x+\left(-\frac{7}{6}\right)^{2}=-\frac{10}{9}+\left(-\frac{7}{6}\right)^{2}$$

Square -\frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=-\frac{10}{9}+\frac{49}{36}$$

Add -\frac{10}{9} to \frac{49}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=0.25$$

Factor x^{2}-\frac{7}{3}x+\frac{49}{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}.

$$\left(x-\frac{7}{6}\right)^{2}=0.25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{0.25}$$

Simplify.

$$x-\frac{7}{6}=\frac{1}{2} x-\frac{7}{6}=-\frac{1}{2}$$

Add \frac{7}{6} to both sides of the equation.

$$x=\frac{5}{3} x=\frac{2}{3}$$