Solve 0=-11/30x^2+1.8x+24

$$0 = - \frac { 11 } { 30 } x ^ { 2 } + 1.8 x + 24$$

Solve for x

$$x=-6$$

$$x = \frac{120}{11} = 10\frac{10}{11} \approx 10.909090909$$

Steps Using the Quadratic Formula

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

$$-\frac{11}{30}x^{2}+1.8x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{11}{30} for a, 1.8 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1.8±\sqrt{1.8^{2}-4\left(-\frac{11}{30}\right)\times 24}}{2\left(-\frac{11}{30}\right)}$$

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

$$x=\frac{-1.8±\sqrt{3.24-4\left(-\frac{11}{30}\right)\times 24}}{2\left(-\frac{11}{30}\right)}$$

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

$$x=\frac{-1.8±\sqrt{3.24+\frac{22}{15}\times 24}}{2\left(-\frac{11}{30}\right)}$$

Multiply \frac{22}{15} times 24.

$$x=\frac{-1.8±\sqrt{3.24+\frac{176}{5}}}{2\left(-\frac{11}{30}\right)}$$

Add 3.24 to \frac{176}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1.8±\sqrt{\frac{961}{25}}}{2\left(-\frac{11}{30}\right)}$$

Take the square root of \frac{961}{25}.

$$x=\frac{-1.8±\frac{31}{5}}{2\left(-\frac{11}{30}\right)}$$

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

$$x=\frac{-1.8±\frac{31}{5}}{-\frac{11}{15}}$$

Now solve the equation x=\frac{-1.8±\frac{31}{5}}{-\frac{11}{15}} when ± is plus. Add -1.8 to \frac{31}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{22}{5}}{-\frac{11}{15}}$$

Divide \frac{22}{5} by -\frac{11}{15} by multiplying \frac{22}{5} by the reciprocal of -\frac{11}{15}.

$$x=-6$$

Now solve the equation x=\frac{-1.8±\frac{31}{5}}{-\frac{11}{15}} when ± is minus. Subtract \frac{31}{5} from -1.8 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{8}{-\frac{11}{15}}$$

Divide -8 by -\frac{11}{15} by multiplying -8 by the reciprocal of -\frac{11}{15}.

$$x=\frac{120}{11}$$

The equation is now solved.

$$x=-6 x=\frac{120}{11}$$

Steps for Completing the Square

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

$$-\frac{11}{30}x^{2}+1.8x+24=0$$

Subtract 24 from both sides. Anything subtracted from zero gives its negation.

$$-\frac{11}{30}x^{2}+1.8x=-24$$

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

$$\frac{-\frac{11}{30}x^{2}+1.8x}{-\frac{11}{30}}=-\frac{24}{-\frac{11}{30}}$$

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

$$x^{2}+\frac{1.8}{-\frac{11}{30}}x=-\frac{24}{-\frac{11}{30}}$$

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

$$x^{2}-\frac{54}{11}x=-\frac{24}{-\frac{11}{30}}$$

Divide -24 by -\frac{11}{30} by multiplying -24 by the reciprocal of -\frac{11}{30}.

$$x^{2}-\frac{54}{11}x=\frac{720}{11}$$

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.

$$x^{2}-\frac{54}{11}x+\left(-\frac{27}{11}\right)^{2}=\frac{720}{11}+\left(-\frac{27}{11}\right)^{2}$$

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

$$x^{2}-\frac{54}{11}x+\frac{729}{121}=\frac{720}{11}+\frac{729}{121}$$

Add \frac{720}{11} to \frac{729}{121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{54}{11}x+\frac{729}{121}=\frac{8649}{121}$$

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}.

$$\left(x-\frac{27}{11}\right)^{2}=\frac{8649}{121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{27}{11}\right)^{2}}=\sqrt{\frac{8649}{121}}$$

Simplify.

$$x-\frac{27}{11}=\frac{93}{11} x-\frac{27}{11}=-\frac{93}{11}$$

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

$$x=\frac{120}{11} x=-6$$