Solve 1.8=-1/5x^2-1.2x+3.2

$$1.8 = - \frac { 1 } { 5 } x ^ { 2 } - 1.2 x + 3.2$$

Solve for x

$$x=-7$$

$$x=1$$

Steps Using the Quadratic Formula

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

$$-\frac{1}{5}x^{2}-1.2x+3.2=1.8$$

Subtract 1.8 from both sides.

$$-\frac{1}{5}x^{2}-1.2x+3.2-1.8=0$$

Subtract 1.8 from 3.2 to get 1.4.

$$-\frac{1}{5}x^{2}-1.2x+1.4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{5} for a, -1.2 for b, and 1.4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1.2\right)±\sqrt{\left(-1.2\right)^{2}-4\left(-\frac{1}{5}\right)\times 1.4}}{2\left(-\frac{1}{5}\right)}$$

Square -1.2 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-1.2\right)±\sqrt{1.44-4\left(-\frac{1}{5}\right)\times 1.4}}{2\left(-\frac{1}{5}\right)}$$

Multiply -4 times -\frac{1}{5}.

$$x=\frac{-\left(-1.2\right)±\sqrt{1.44+\frac{4}{5}\times 1.4}}{2\left(-\frac{1}{5}\right)}$$

Multiply \frac{4}{5} times 1.4 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-1.2\right)±\sqrt{\frac{36+28}{25}}}{2\left(-\frac{1}{5}\right)}$$

Add 1.44 to \frac{28}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-1.2\right)±\sqrt{\frac{64}{25}}}{2\left(-\frac{1}{5}\right)}$$

Take the square root of \frac{64}{25}.

$$x=\frac{-\left(-1.2\right)±\frac{8}{5}}{2\left(-\frac{1}{5}\right)}$$

The opposite of -1.2 is 1.2.

$$x=\frac{1.2±\frac{8}{5}}{2\left(-\frac{1}{5}\right)}$$

Multiply 2 times -\frac{1}{5}.

$$x=\frac{1.2±\frac{8}{5}}{-\frac{2}{5}}$$

Now solve the equation x=\frac{1.2±\frac{8}{5}}{-\frac{2}{5}} when ± is plus. Add 1.2 to \frac{8}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{14}{5}}{-\frac{2}{5}}$$

Divide \frac{14}{5} by -\frac{2}{5} by multiplying \frac{14}{5} by the reciprocal of -\frac{2}{5}.

$$x=-7$$

Now solve the equation x=\frac{1.2±\frac{8}{5}}{-\frac{2}{5}} when ± is minus. Subtract \frac{8}{5} from 1.2 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide -\frac{2}{5} by -\frac{2}{5} by multiplying -\frac{2}{5} by the reciprocal of -\frac{2}{5}.

$$x=1$$

The equation is now solved.

$$x=-7 x=1$$

Steps for Completing the Square

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

$$-\frac{1}{5}x^{2}-1.2x+3.2=1.8$$

Subtract 3.2 from both sides.

$$-\frac{1}{5}x^{2}-1.2x=1.8-3.2$$

Subtract 3.2 from 1.8 to get -1.4.

$$-\frac{1}{5}x^{2}-1.2x=-1.4$$

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.

$$-\frac{1}{5}x^{2}-1.2x=-\frac{7}{5}$$

Multiply both sides by -5.

$$\frac{-\frac{1}{5}x^{2}-1.2x}{-\frac{1}{5}}=-\frac{\frac{7}{5}}{-\frac{1}{5}}$$

Dividing by -\frac{1}{5} undoes the multiplication by -\frac{1}{5}.

$$x^{2}+\left(-\frac{1.2}{-\frac{1}{5}}\right)x=-\frac{\frac{7}{5}}{-\frac{1}{5}}$$

Divide -1.2 by -\frac{1}{5} by multiplying -1.2 by the reciprocal of -\frac{1}{5}.

$$x^{2}+6x=-\frac{\frac{7}{5}}{-\frac{1}{5}}$$

Divide -\frac{7}{5} by -\frac{1}{5} by multiplying -\frac{7}{5} by the reciprocal of -\frac{1}{5}.

$$x^{2}+6x=7$$

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

$$x^{2}+6x+3^{2}=7+3^{2}$$

Square 3.

$$x^{2}+6x+9=7+9$$

Add 7 to 9.

$$x^{2}+6x+9=16$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{16}$$

Simplify.

$$x+3=4 x+3=-4$$

Subtract 3 from both sides of the equation.

$$x=1 x=-7$$