Solve -3x^2+408x-1800=2

$$-3 { x }^{ 2 } +408x-1800 = 2$$

Solve for x

$$x=\frac{\sqrt{36210}}{3}+68\approx 131.429751169$$

$$x=-\frac{\sqrt{36210}}{3}+68\approx 4.570248831$$

Steps Using the Quadratic Formula

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.

$$-3x^{2}+408x-1800=2$$

Subtract 2 from both sides of the equation.

$$-3x^{2}+408x-1800-2=2-2$$

Subtracting 2 from itself leaves 0.

$$-3x^{2}+408x-1800-2=0$$

Subtract 2 from -1800.

$$-3x^{2}+408x-1802=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 408 for b, and -1802 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-408±\sqrt{408^{2}-4\left(-3\right)\left(-1802\right)}}{2\left(-3\right)}$$

Square 408.

$$x=\frac{-408±\sqrt{166464-4\left(-3\right)\left(-1802\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-408±\sqrt{166464+12\left(-1802\right)}}{2\left(-3\right)}$$

Multiply 12 times -1802.

$$x=\frac{-408±\sqrt{166464-21624}}{2\left(-3\right)}$$

Add 166464 to -21624.

$$x=\frac{-408±\sqrt{144840}}{2\left(-3\right)}$$

Take the square root of 144840.

$$x=\frac{-408±2\sqrt{36210}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-408±2\sqrt{36210}}{-6}$$

Now solve the equation x=\frac{-408±2\sqrt{36210}}{-6} when ± is plus. Add -408 to 2\sqrt{36210}.

$$x=\frac{2\sqrt{36210}-408}{-6}$$

Divide -408+2\sqrt{36210} by -6.

$$x=-\frac{\sqrt{36210}}{3}+68$$

Now solve the equation x=\frac{-408±2\sqrt{36210}}{-6} when ± is minus. Subtract 2\sqrt{36210} from -408.

$$x=\frac{-2\sqrt{36210}-408}{-6}$$

Divide -408-2\sqrt{36210} by -6.

$$x=\frac{\sqrt{36210}}{3}+68$$

The equation is now solved.

$$x=-\frac{\sqrt{36210}}{3}+68 x=\frac{\sqrt{36210}}{3}+68$$

Steps for Completing the Square

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.

$$-3x^{2}+408x-1800=2$$

Add 1800 to both sides of the equation.

$$-3x^{2}+408x-1800-\left(-1800\right)=2-\left(-1800\right)$$

Subtracting -1800 from itself leaves 0.

$$-3x^{2}+408x=2-\left(-1800\right)$$

Subtract -1800 from 2.

$$-3x^{2}+408x=1802$$

Divide both sides by -3.

$$\frac{-3x^{2}+408x}{-3}=\frac{1802}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{408}{-3}x=\frac{1802}{-3}$$

Divide 408 by -3.

$$x^{2}-136x=\frac{1802}{-3}$$

Divide 1802 by -3.

$$x^{2}-136x=-\frac{1802}{3}$$

Divide -136, the coefficient of the x term, by 2 to get -68. Then add the square of -68 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-136x+\left(-68\right)^{2}=-\frac{1802}{3}+\left(-68\right)^{2}$$

Square -68.

$$x^{2}-136x+4624=-\frac{1802}{3}+4624$$

Add -\frac{1802}{3} to 4624.

$$x^{2}-136x+4624=\frac{12070}{3}$$

Factor x^{2}-136x+4624. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-68\right)^{2}=\frac{12070}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-68\right)^{2}}=\sqrt{\frac{12070}{3}}$$

Simplify.

$$x-68=\frac{\sqrt{36210}}{3} x-68=-\frac{\sqrt{36210}}{3}$$

Add 68 to both sides of the equation.

$$x=\frac{\sqrt{36210}}{3}+68 x=-\frac{\sqrt{36210}}{3}+68$$