Solve 2x^2-90x-3600=0

$$2 { x }^{ 2 } -90x-3600=0$$

Solve for x

$$x = \frac{15 \sqrt{41} + 45}{2} \approx 70.523431781$$

$$x=\frac{45-15\sqrt{41}}{2}\approx -25.523431781$$

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.

$$2x^{2}-90x-3600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -90 for b, and -3600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 2\left(-3600\right)}}{2\times 2}$$

Square -90.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\times 2\left(-3600\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-90\right)±\sqrt{8100-8\left(-3600\right)}}{2\times 2}$$

Multiply -8 times -3600.

$$x=\frac{-\left(-90\right)±\sqrt{8100+28800}}{2\times 2}$$

Add 8100 to 28800.

$$x=\frac{-\left(-90\right)±\sqrt{36900}}{2\times 2}$$

Take the square root of 36900.

$$x=\frac{-\left(-90\right)±30\sqrt{41}}{2\times 2}$$

The opposite of -90 is 90.

$$x=\frac{90±30\sqrt{41}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{90±30\sqrt{41}}{4}$$

Now solve the equation x=\frac{90±30\sqrt{41}}{4} when ± is plus. Add 90 to 30\sqrt{41}.

$$x=\frac{30\sqrt{41}+90}{4}$$

Divide 90+30\sqrt{41} by 4.

$$x=\frac{15\sqrt{41}+45}{2}$$

Now solve the equation x=\frac{90±30\sqrt{41}}{4} when ± is minus. Subtract 30\sqrt{41} from 90.

$$x=\frac{90-30\sqrt{41}}{4}$$

Divide 90-30\sqrt{41} by 4.

$$x=\frac{45-15\sqrt{41}}{2}$$

The equation is now solved.

$$x=\frac{15\sqrt{41}+45}{2} x=\frac{45-15\sqrt{41}}{2}$$

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.

$$2x^{2}-90x-3600=0$$

Add 3600 to both sides of the equation.

$$2x^{2}-90x-3600-\left(-3600\right)=-\left(-3600\right)$$

Subtracting -3600 from itself leaves 0.

$$2x^{2}-90x=-\left(-3600\right)$$

Subtract -3600 from 0.

$$2x^{2}-90x=3600$$

Divide both sides by 2.

$$\frac{2x^{2}-90x}{2}=\frac{3600}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{90}{2}\right)x=\frac{3600}{2}$$

Divide -90 by 2.

$$x^{2}-45x=\frac{3600}{2}$$

Divide 3600 by 2.

$$x^{2}-45x=1800$$

Divide -45, the coefficient of the x term, by 2 to get -\frac{45}{2}. Then add the square of -\frac{45}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-45x+\left(-\frac{45}{2}\right)^{2}=1800+\left(-\frac{45}{2}\right)^{2}$$

Square -\frac{45}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-45x+\frac{2025}{4}=1800+\frac{2025}{4}$$

Add 1800 to \frac{2025}{4}.

$$x^{2}-45x+\frac{2025}{4}=\frac{9225}{4}$$

Factor x^{2}-45x+\frac{2025}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{45}{2}\right)^{2}=\frac{9225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{45}{2}\right)^{2}}=\sqrt{\frac{9225}{4}}$$

Simplify.

$$x-\frac{45}{2}=\frac{15\sqrt{41}}{2} x-\frac{45}{2}=-\frac{15\sqrt{41}}{2}$$

Add \frac{45}{2} to both sides of the equation.

$$x=\frac{15\sqrt{41}+45}{2} x=\frac{45-15\sqrt{41}}{2}$$