Solve =1.8=-11/30x^2+1.8x+2.4

$$= 1.8 = - \frac { 11 } { 30 } x ^ { 2 } + 1.8 x + 2.4$$

Solve for x

$$x = \frac{3 \sqrt{103} + 27}{11} \approx 5.222424972$$

$$x=\frac{27-3\sqrt{103}}{11}\approx -0.313334063$$

Steps Using the Quadratic Formula

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

$$-\frac{11}{30}x^{2}+1.8x+2.4=1.8$$

Subtract 1.8 from both sides.

$$-\frac{11}{30}x^{2}+1.8x+2.4-1.8=0$$

Subtract 1.8 from 2.4 to get 0.6.

$$-\frac{11}{30}x^{2}+1.8x+0.6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{11}{30} for a, 1.8 for b, and 0.6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1.8±\sqrt{1.8^{2}-4\left(-\frac{11}{30}\right)\times 0.6}}{2\left(-\frac{11}{30}\right)}$$

Square 1.8 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-1.8±\sqrt{3.24-4\left(-\frac{11}{30}\right)\times 0.6}}{2\left(-\frac{11}{30}\right)}$$

Multiply -4 times -\frac{11}{30}.

$$x=\frac{-1.8±\sqrt{3.24+\frac{22}{15}\times 0.6}}{2\left(-\frac{11}{30}\right)}$$

Multiply \frac{22}{15} times 0.6 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1.8±\sqrt{\frac{81+22}{25}}}{2\left(-\frac{11}{30}\right)}$$

Add 3.24 to \frac{22}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1.8±\sqrt{\frac{103}{25}}}{2\left(-\frac{11}{30}\right)}$$

Take the square root of \frac{103}{25}.

$$x=\frac{-1.8±\frac{\sqrt{103}}{5}}{2\left(-\frac{11}{30}\right)}$$

Multiply 2 times -\frac{11}{30}.

$$x=\frac{-1.8±\frac{\sqrt{103}}{5}}{-\frac{11}{15}}$$

Now solve the equation x=\frac{-1.8±\frac{\sqrt{103}}{5}}{-\frac{11}{15}} when ± is plus. Add -1.8 to \frac{\sqrt{103}}{5}.

$$x=\frac{\sqrt{103}-9}{-\frac{11}{15}\times 5}$$

Divide \frac{-9+\sqrt{103}}{5} by -\frac{11}{15} by multiplying \frac{-9+\sqrt{103}}{5} by the reciprocal of -\frac{11}{15}.

$$x=\frac{27-3\sqrt{103}}{11}$$

Now solve the equation x=\frac{-1.8±\frac{\sqrt{103}}{5}}{-\frac{11}{15}} when ± is minus. Subtract \frac{\sqrt{103}}{5} from -1.8.

$$x=\frac{-\sqrt{103}-9}{-\frac{11}{15}\times 5}$$

Divide \frac{-9-\sqrt{103}}{5} by -\frac{11}{15} by multiplying \frac{-9-\sqrt{103}}{5} by the reciprocal of -\frac{11}{15}.

$$x=\frac{3\sqrt{103}+27}{11}$$

The equation is now solved.

$$x=\frac{27-3\sqrt{103}}{11} x=\frac{3\sqrt{103}+27}{11}$$

Steps for Completing the Square

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

$$-\frac{11}{30}x^{2}+1.8x+2.4=1.8$$

Subtract 2.4 from both sides.

$$-\frac{11}{30}x^{2}+1.8x=1.8-2.4$$

Subtract 2.4 from 1.8 to get -0.6.

$$-\frac{11}{30}x^{2}+1.8x=-0.6$$

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.

$$-\frac{11}{30}x^{2}+1.8x=-\frac{3}{5}$$

Divide both sides of the equation by -\frac{11}{30}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{11}{30}x^{2}+1.8x}{-\frac{11}{30}}=-\frac{\frac{3}{5}}{-\frac{11}{30}}$$

Dividing by -\frac{11}{30} undoes the multiplication by -\frac{11}{30}.

$$x^{2}+\frac{1.8}{-\frac{11}{30}}x=-\frac{\frac{3}{5}}{-\frac{11}{30}}$$

Divide 1.8 by -\frac{11}{30} by multiplying 1.8 by the reciprocal of -\frac{11}{30}.

$$x^{2}-\frac{54}{11}x=-\frac{\frac{3}{5}}{-\frac{11}{30}}$$

Divide -\frac{3}{5} by -\frac{11}{30} by multiplying -\frac{3}{5} by the reciprocal of -\frac{11}{30}.

$$x^{2}-\frac{54}{11}x=\frac{18}{11}$$

Divide -\frac{54}{11}, the coefficient of the x term, by 2 to get -\frac{27}{11}. Then add the square of -\frac{27}{11} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{54}{11}x+\left(-\frac{27}{11}\right)^{2}=\frac{18}{11}+\left(-\frac{27}{11}\right)^{2}$$

Square -\frac{27}{11} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{54}{11}x+\frac{729}{121}=\frac{18}{11}+\frac{729}{121}$$

Add \frac{18}{11} to \frac{729}{121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{54}{11}x+\frac{729}{121}=\frac{927}{121}$$

Factor x^{2}-\frac{54}{11}x+\frac{729}{121}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{27}{11}\right)^{2}=\frac{927}{121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{27}{11}\right)^{2}}=\sqrt{\frac{927}{121}}$$

Simplify.

$$x-\frac{27}{11}=\frac{3\sqrt{103}}{11} x-\frac{27}{11}=-\frac{3\sqrt{103}}{11}$$

Add \frac{27}{11} to both sides of the equation.

$$x=\frac{3\sqrt{103}+27}{11} x=\frac{27-3\sqrt{103}}{11}$$