Solve 2400=-10x^2+1140x-29600

$$2400=-10 { x }^{ 2 } +1140x-29600$$

Solve for x

$$x=50$$

$$x=64$$

Steps Using the Quadratic Formula

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

$$-10x^{2}+1140x-29600=2400$$

Subtract 2400 from both sides.

$$-10x^{2}+1140x-29600-2400=0$$

Subtract 2400 from -29600 to get -32000.

$$-10x^{2}+1140x-32000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 1140 for b, and -32000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1140±\sqrt{1140^{2}-4\left(-10\right)\left(-32000\right)}}{2\left(-10\right)}$$

Square 1140.

$$x=\frac{-1140±\sqrt{1299600-4\left(-10\right)\left(-32000\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-1140±\sqrt{1299600+40\left(-32000\right)}}{2\left(-10\right)}$$

Multiply 40 times -32000.

$$x=\frac{-1140±\sqrt{1299600-1280000}}{2\left(-10\right)}$$

Add 1299600 to -1280000.

$$x=\frac{-1140±\sqrt{19600}}{2\left(-10\right)}$$

Take the square root of 19600.

$$x=\frac{-1140±140}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-1140±140}{-20}$$

Now solve the equation x=\frac{-1140±140}{-20} when ± is plus. Add -1140 to 140.

$$x=-\frac{1000}{-20}$$

Divide -1000 by -20.

$$x=50$$

Now solve the equation x=\frac{-1140±140}{-20} when ± is minus. Subtract 140 from -1140.

$$x=-\frac{1280}{-20}$$

Divide -1280 by -20.

$$x=64$$

The equation is now solved.

$$x=50 x=64$$

Steps for Completing the Square

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

$$-10x^{2}+1140x-29600=2400$$

Add 29600 to both sides.

$$-10x^{2}+1140x=2400+29600$$

Add 2400 and 29600 to get 32000.

$$-10x^{2}+1140x=32000$$

Divide both sides by -10.

$$\frac{-10x^{2}+1140x}{-10}=\frac{32000}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{1140}{-10}x=\frac{32000}{-10}$$

Divide 1140 by -10.

$$x^{2}-114x=\frac{32000}{-10}$$

Divide 32000 by -10.

$$x^{2}-114x=-3200$$

Divide -114, the coefficient of the x term, by 2 to get -57. Then add the square of -57 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-114x+\left(-57\right)^{2}=-3200+\left(-57\right)^{2}$$

Square -57.

$$x^{2}-114x+3249=-3200+3249$$

Add -3200 to 3249.

$$x^{2}-114x+3249=49$$

Factor x^{2}-114x+3249. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-57\right)^{2}=49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-57\right)^{2}}=\sqrt{49}$$

Simplify.

$$x-57=7 x-57=-7$$

Add 57 to both sides of the equation.

$$x=64 x=50$$