Solve 18x-1.5x^2+16=-600x

$$18x-1.5 { x }^{ 2 } +16 = -600x$$

Solve for x

$$x=\frac{2\sqrt{95505}}{3}+206\approx 412.025888341$$

$$x=-\frac{2\sqrt{95505}}{3}+206\approx -0.025888341$$

Steps Using the Quadratic Formula

Add 600x to both sides.

$$18x-1.5x^{2}+16+600x=0$$

Combine 18x and 600x to get 618x.

$$618x-1.5x^{2}+16=0$$

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.

$$-1.5x^{2}+618x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1.5 for a, 618 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-618±\sqrt{618^{2}-4\left(-1.5\right)\times 16}}{2\left(-1.5\right)}$$

Square 618.

$$x=\frac{-618±\sqrt{381924-4\left(-1.5\right)\times 16}}{2\left(-1.5\right)}$$

Multiply -4 times -1.5.

$$x=\frac{-618±\sqrt{381924+6\times 16}}{2\left(-1.5\right)}$$

Multiply 6 times 16.

$$x=\frac{-618±\sqrt{381924+96}}{2\left(-1.5\right)}$$

Add 381924 to 96.

$$x=\frac{-618±\sqrt{382020}}{2\left(-1.5\right)}$$

Take the square root of 382020.

$$x=\frac{-618±2\sqrt{95505}}{2\left(-1.5\right)}$$

Multiply 2 times -1.5.

$$x=\frac{-618±2\sqrt{95505}}{-3}$$

Now solve the equation x=\frac{-618±2\sqrt{95505}}{-3} when ± is plus. Add -618 to 2\sqrt{95505}.

$$x=\frac{2\sqrt{95505}-618}{-3}$$

Divide -618+2\sqrt{95505} by -3.

$$x=-\frac{2\sqrt{95505}}{3}+206$$

Now solve the equation x=\frac{-618±2\sqrt{95505}}{-3} when ± is minus. Subtract 2\sqrt{95505} from -618.

$$x=\frac{-2\sqrt{95505}-618}{-3}$$

Divide -618-2\sqrt{95505} by -3.

$$x=\frac{2\sqrt{95505}}{3}+206$$

The equation is now solved.

$$x=-\frac{2\sqrt{95505}}{3}+206 x=\frac{2\sqrt{95505}}{3}+206$$

Steps for Completing the Square

Add 600x to both sides.

$$18x-1.5x^{2}+16+600x=0$$

Combine 18x and 600x to get 618x.

$$618x-1.5x^{2}+16=0$$

Subtract 16 from both sides. Anything subtracted from zero gives its negation.

$$618x-1.5x^{2}=-16$$

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.

$$-1.5x^{2}+618x=-16$$

Divide both sides of the equation by -1.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-1.5x^{2}+618x}{-1.5}=-\frac{16}{-1.5}$$

Dividing by -1.5 undoes the multiplication by -1.5.

$$x^{2}+\frac{618}{-1.5}x=-\frac{16}{-1.5}$$

Divide 618 by -1.5 by multiplying 618 by the reciprocal of -1.5.

$$x^{2}-412x=-\frac{16}{-1.5}$$

Divide -16 by -1.5 by multiplying -16 by the reciprocal of -1.5.

$$x^{2}-412x=\frac{32}{3}$$

Divide -412, the coefficient of the x term, by 2 to get -206. Then add the square of -206 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-412x+\left(-206\right)^{2}=\frac{32}{3}+\left(-206\right)^{2}$$

Square -206.

$$x^{2}-412x+42436=\frac{32}{3}+42436$$

Add \frac{32}{3} to 42436.

$$x^{2}-412x+42436=\frac{127340}{3}$$

Factor x^{2}-412x+42436. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-206\right)^{2}=\frac{127340}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-206\right)^{2}}=\sqrt{\frac{127340}{3}}$$

Simplify.

$$x-206=\frac{2\sqrt{95505}}{3} x-206=-\frac{2\sqrt{95505}}{3}$$

Add 206 to both sides of the equation.

$$x=\frac{2\sqrt{95505}}{3}+206 x=-\frac{2\sqrt{95505}}{3}+206$$