Solve 2x^2+3000x-35000=0

$$2 { x }^{ 2 } +3000x-35000=0$$

Solve for x

$$x=100\sqrt{58}-750\approx 11.577310586$$

$$x=-100\sqrt{58}-750\approx -1511.577310586$$

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.

$$2x^{2}+3000x-35000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 3000 for b, and -35000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3000±\sqrt{3000^{2}-4\times 2\left(-35000\right)}}{2\times 2}$$

Square 3000.

$$x=\frac{-3000±\sqrt{9000000-4\times 2\left(-35000\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-3000±\sqrt{9000000-8\left(-35000\right)}}{2\times 2}$$

Multiply -8 times -35000.

$$x=\frac{-3000±\sqrt{9000000+280000}}{2\times 2}$$

Add 9000000 to 280000.

$$x=\frac{-3000±\sqrt{9280000}}{2\times 2}$$

Take the square root of 9280000.

$$x=\frac{-3000±400\sqrt{58}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-3000±400\sqrt{58}}{4}$$

Now solve the equation x=\frac{-3000±400\sqrt{58}}{4} when ± is plus. Add -3000 to 400\sqrt{58}.

$$x=\frac{400\sqrt{58}-3000}{4}$$

Divide -3000+400\sqrt{58} by 4.

$$x=100\sqrt{58}-750$$

Now solve the equation x=\frac{-3000±400\sqrt{58}}{4} when ± is minus. Subtract 400\sqrt{58} from -3000.

$$x=\frac{-400\sqrt{58}-3000}{4}$$

Divide -3000-400\sqrt{58} by 4.

$$x=-100\sqrt{58}-750$$

The equation is now solved.

$$x=100\sqrt{58}-750 x=-100\sqrt{58}-750$$

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.

$$2x^{2}+3000x-35000=0$$

Add 35000 to both sides of the equation.

$$2x^{2}+3000x-35000-\left(-35000\right)=-\left(-35000\right)$$

Subtracting -35000 from itself leaves 0.

$$2x^{2}+3000x=-\left(-35000\right)$$

Subtract -35000 from 0.

$$2x^{2}+3000x=35000$$

Divide both sides by 2.

$$\frac{2x^{2}+3000x}{2}=\frac{35000}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{3000}{2}x=\frac{35000}{2}$$

Divide 3000 by 2.

$$x^{2}+1500x=\frac{35000}{2}$$

Divide 35000 by 2.

$$x^{2}+1500x=17500$$

Divide 1500, the coefficient of the x term, by 2 to get 750. Then add the square of 750 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+1500x+750^{2}=17500+750^{2}$$

Square 750.

$$x^{2}+1500x+562500=17500+562500$$

Add 17500 to 562500.

$$x^{2}+1500x+562500=580000$$

Factor x^{2}+1500x+562500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+750\right)^{2}=580000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+750\right)^{2}}=\sqrt{580000}$$

Simplify.

$$x+750=100\sqrt{58} x+750=-100\sqrt{58}$$

Subtract 750 from both sides of the equation.

$$x=100\sqrt{58}-750 x=-100\sqrt{58}-750$$