Solve 0=-0.02x^2+8x+1.8

$$0 = -0.02 { x }^{ 2 } +8x+1.8$$

Solve for x

$$x=\sqrt{40090}+200\approx 400.22487358$$

$$x=200-\sqrt{40090}\approx -0.22487358$$

Steps Using the Quadratic Formula

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

$$-0.02x^{2}+8x+1.8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.02 for a, 8 for b, and 1.8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-0.02\right)\times 1.8}}{2\left(-0.02\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-0.02\right)\times 1.8}}{2\left(-0.02\right)}$$

Multiply -4 times -0.02.

$$x=\frac{-8±\sqrt{64+0.08\times 1.8}}{2\left(-0.02\right)}$$

Multiply 0.08 times 1.8 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-8±\sqrt{64+0.144}}{2\left(-0.02\right)}$$

Add 64 to 0.144.

$$x=\frac{-8±\sqrt{64.144}}{2\left(-0.02\right)}$$

Take the square root of 64.144.

$$x=\frac{-8±\frac{\sqrt{40090}}{25}}{2\left(-0.02\right)}$$

Multiply 2 times -0.02.

$$x=\frac{-8±\frac{\sqrt{40090}}{25}}{-0.04}$$

Now solve the equation x=\frac{-8±\frac{\sqrt{40090}}{25}}{-0.04} when ± is plus. Add -8 to \frac{\sqrt{40090}}{25}.

$$x=\frac{\frac{\sqrt{40090}}{25}-8}{-0.04}$$

Divide -8+\frac{\sqrt{40090}}{25} by -0.04 by multiplying -8+\frac{\sqrt{40090}}{25} by the reciprocal of -0.04.

$$x=200-\sqrt{40090}$$

Now solve the equation x=\frac{-8±\frac{\sqrt{40090}}{25}}{-0.04} when ± is minus. Subtract \frac{\sqrt{40090}}{25} from -8.

$$x=\frac{-\frac{\sqrt{40090}}{25}-8}{-0.04}$$

Divide -8-\frac{\sqrt{40090}}{25} by -0.04 by multiplying -8-\frac{\sqrt{40090}}{25} by the reciprocal of -0.04.

$$x=\sqrt{40090}+200$$

The equation is now solved.

$$x=200-\sqrt{40090} x=\sqrt{40090}+200$$

Steps for Completing the Square

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

$$-0.02x^{2}+8x+1.8=0$$

Subtract 1.8 from both sides. Anything subtracted from zero gives its negation.

$$-0.02x^{2}+8x=-1.8$$

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.

$$-0.02x^{2}+8x=-\frac{9}{5}$$

Multiply both sides by -50.

$$\frac{-0.02x^{2}+8x}{-0.02}=-\frac{\frac{9}{5}}{-0.02}$$

Dividing by -0.02 undoes the multiplication by -0.02.

$$x^{2}+\frac{8}{-0.02}x=-\frac{\frac{9}{5}}{-0.02}$$

Divide 8 by -0.02 by multiplying 8 by the reciprocal of -0.02.

$$x^{2}-400x=-\frac{\frac{9}{5}}{-0.02}$$

Divide -\frac{9}{5} by -0.02 by multiplying -\frac{9}{5} by the reciprocal of -0.02.

$$x^{2}-400x=90$$

Divide -400, the coefficient of the x term, by 2 to get -200. Then add the square of -200 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-400x+\left(-200\right)^{2}=90+\left(-200\right)^{2}$$

Square -200.

$$x^{2}-400x+40000=90+40000$$

Add 90 to 40000.

$$x^{2}-400x+40000=40090$$

Factor x^{2}-400x+40000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-200\right)^{2}=40090$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-200\right)^{2}}=\sqrt{40090}$$

Simplify.

$$x-200=\sqrt{40090} x-200=-\sqrt{40090}$$

Add 200 to both sides of the equation.

$$x=\sqrt{40090}+200 x=200-\sqrt{40090}$$