Solve 0.12=6.1803x+4.905x^2

$$0.12=6.1803x+4.905 { x }^{ 2 }$$

Solve for x

$$x=\frac{\sqrt{450561201}}{32700}-0.63\approx 0.019126206$$

$$x=-\frac{\sqrt{450561201}}{32700}-0.63\approx -1.279126206$$

Steps Using the Quadratic Formula

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

$$6.1803x+4.905x^{2}=0.12$$

Subtract 0.12 from both sides.

$$6.1803x+4.905x^{2}-0.12=0$$

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.

$$4.905x^{2}+6.1803x-0.12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4.905 for a, 6.1803 for b, and -0.12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6.1803±\sqrt{6.1803^{2}-4\times 4.905\left(-0.12\right)}}{2\times 4.905}$$

Square 6.1803 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-6.1803±\sqrt{38.19610809-4\times 4.905\left(-0.12\right)}}{2\times 4.905}$$

Multiply -4 times 4.905.

$$x=\frac{-6.1803±\sqrt{38.19610809-19.62\left(-0.12\right)}}{2\times 4.905}$$

Multiply -19.62 times -0.12 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-6.1803±\sqrt{38.19610809+2.3544}}{2\times 4.905}$$

Add 38.19610809 to 2.3544 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-6.1803±\sqrt{40.55050809}}{2\times 4.905}$$

Take the square root of 40.55050809.

$$x=\frac{-6.1803±\frac{3\sqrt{450561201}}{10000}}{2\times 4.905}$$

Multiply 2 times 4.905.

$$x=\frac{-6.1803±\frac{3\sqrt{450561201}}{10000}}{9.81}$$

Now solve the equation x=\frac{-6.1803±\frac{3\sqrt{450561201}}{10000}}{9.81} when ± is plus. Add -6.1803 to \frac{3\sqrt{450561201}}{10000}.

$$x=\frac{3\sqrt{450561201}-61803}{9.81\times 10000}$$

Divide \frac{-61803+3\sqrt{450561201}}{10000} by 9.81 by multiplying \frac{-61803+3\sqrt{450561201}}{10000} by the reciprocal of 9.81.

$$x=\frac{\sqrt{450561201}}{32700}-\frac{63}{100}$$

Now solve the equation x=\frac{-6.1803±\frac{3\sqrt{450561201}}{10000}}{9.81} when ± is minus. Subtract \frac{3\sqrt{450561201}}{10000} from -6.1803.

$$x=\frac{-3\sqrt{450561201}-61803}{9.81\times 10000}$$

Divide \frac{-61803-3\sqrt{450561201}}{10000} by 9.81 by multiplying \frac{-61803-3\sqrt{450561201}}{10000} by the reciprocal of 9.81.

$$x=-\frac{\sqrt{450561201}}{32700}-\frac{63}{100}$$

The equation is now solved.

$$x=\frac{\sqrt{450561201}}{32700}-\frac{63}{100} x=-\frac{\sqrt{450561201}}{32700}-\frac{63}{100}$$

Steps for Completing the Square

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

$$6.1803x+4.905x^{2}=0.12$$

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.

$$4.905x^{2}+6.1803x=0.12$$

Divide both sides of the equation by 4.905, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{4.905x^{2}+6.1803x}{4.905}=\frac{0.12}{4.905}$$

Dividing by 4.905 undoes the multiplication by 4.905.

$$x^{2}+\frac{6.1803}{4.905}x=\frac{0.12}{4.905}$$

Divide 6.1803 by 4.905 by multiplying 6.1803 by the reciprocal of 4.905.

$$x^{2}+1.26x=\frac{0.12}{4.905}$$

Divide 0.12 by 4.905 by multiplying 0.12 by the reciprocal of 4.905.

$$x^{2}+1.26x=\frac{8}{327}$$

Divide 1.26, the coefficient of the x term, by 2 to get 0.63. Then add the square of 0.63 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+1.26x+0.63^{2}=\frac{8}{327}+0.63^{2}$$

Square 0.63 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+1.26x+0.3969=\frac{8}{327}+0.3969$$

Add \frac{8}{327} to 0.3969 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+1.26x+0.3969=\frac{1377863}{3270000}$$

Factor x^{2}+1.26x+0.3969. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+0.63\right)^{2}=\frac{1377863}{3270000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+0.63\right)^{2}}=\sqrt{\frac{1377863}{3270000}}$$

Simplify.

$$x+0.63=\frac{\sqrt{450561201}}{32700} x+0.63=-\frac{\sqrt{450561201}}{32700}$$

Subtract 0.63 from both sides of the equation.

$$x=\frac{\sqrt{450561201}}{32700}-\frac{63}{100} x=-\frac{\sqrt{450561201}}{32700}-\frac{63}{100}$$