Solve 2x^2-7500+300x=0

$$2 { x }^{ 2 } -7500+300x=0$$

Solve for x

$$x=25\sqrt{15}-75\approx 21.824583655$$

$$x=-25\sqrt{15}-75\approx -171.824583655$$

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.

$$2x^{2}+300x-7500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 300 for b, and -7500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-300±\sqrt{300^{2}-4\times 2\left(-7500\right)}}{2\times 2}$$

Square 300.

$$x=\frac{-300±\sqrt{90000-4\times 2\left(-7500\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-300±\sqrt{90000-8\left(-7500\right)}}{2\times 2}$$

Multiply -8 times -7500.

$$x=\frac{-300±\sqrt{90000+60000}}{2\times 2}$$

Add 90000 to 60000.

$$x=\frac{-300±\sqrt{150000}}{2\times 2}$$

Take the square root of 150000.

$$x=\frac{-300±100\sqrt{15}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-300±100\sqrt{15}}{4}$$

Now solve the equation x=\frac{-300±100\sqrt{15}}{4} when ± is plus. Add -300 to 100\sqrt{15}.

$$x=\frac{100\sqrt{15}-300}{4}$$

Divide -300+100\sqrt{15} by 4.

$$x=25\sqrt{15}-75$$

Now solve the equation x=\frac{-300±100\sqrt{15}}{4} when ± is minus. Subtract 100\sqrt{15} from -300.

$$x=\frac{-100\sqrt{15}-300}{4}$$

Divide -300-100\sqrt{15} by 4.

$$x=-25\sqrt{15}-75$$

The equation is now solved.

$$x=25\sqrt{15}-75 x=-25\sqrt{15}-75$$

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.

$$2x^{2}+300x-7500=0$$

Add 7500 to both sides of the equation.

$$2x^{2}+300x-7500-\left(-7500\right)=-\left(-7500\right)$$

Subtracting -7500 from itself leaves 0.

$$2x^{2}+300x=-\left(-7500\right)$$

Subtract -7500 from 0.

$$2x^{2}+300x=7500$$

Divide both sides by 2.

$$\frac{2x^{2}+300x}{2}=\frac{7500}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{300}{2}x=\frac{7500}{2}$$

Divide 300 by 2.

$$x^{2}+150x=\frac{7500}{2}$$

Divide 7500 by 2.

$$x^{2}+150x=3750$$

Divide 150, the coefficient of the x term, by 2 to get 75. Then add the square of 75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+150x+75^{2}=3750+75^{2}$$

Square 75.

$$x^{2}+150x+5625=3750+5625$$

Add 3750 to 5625.

$$x^{2}+150x+5625=9375$$

Factor x^{2}+150x+5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+75\right)^{2}=9375$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+75\right)^{2}}=\sqrt{9375}$$

Simplify.

$$x+75=25\sqrt{15} x+75=-25\sqrt{15}$$

Subtract 75 from both sides of the equation.

$$x=25\sqrt{15}-75 x=-25\sqrt{15}-75$$