Solve x^2-810x+7800=0

$${ x }^{ 2 } -810x+7800 = 0$$

Solve for x

$$x=5\sqrt{6249}+405\approx 800.253083479$$

$$x=405-5\sqrt{6249}\approx 9.746916521$$

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.

$$x^{2}-810x+7800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -810 for b, and 7800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-810\right)±\sqrt{\left(-810\right)^{2}-4\times 7800}}{2}$$

Square -810.

$$x=\frac{-\left(-810\right)±\sqrt{656100-4\times 7800}}{2}$$

Multiply -4 times 7800.

$$x=\frac{-\left(-810\right)±\sqrt{656100-31200}}{2}$$

Add 656100 to -31200.

$$x=\frac{-\left(-810\right)±\sqrt{624900}}{2}$$

Take the square root of 624900.

$$x=\frac{-\left(-810\right)±10\sqrt{6249}}{2}$$

The opposite of -810 is 810.

$$x=\frac{810±10\sqrt{6249}}{2}$$

Now solve the equation x=\frac{810±10\sqrt{6249}}{2} when ± is plus. Add 810 to 10\sqrt{6249}.

$$x=\frac{10\sqrt{6249}+810}{2}$$

Divide 810+10\sqrt{6249} by 2.

$$x=5\sqrt{6249}+405$$

Now solve the equation x=\frac{810±10\sqrt{6249}}{2} when ± is minus. Subtract 10\sqrt{6249} from 810.

$$x=\frac{810-10\sqrt{6249}}{2}$$

Divide 810-10\sqrt{6249} by 2.

$$x=405-5\sqrt{6249}$$

The equation is now solved.

$$x=5\sqrt{6249}+405 x=405-5\sqrt{6249}$$

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.

$$x^{2}-810x+7800=0$$

Subtract 7800 from both sides of the equation.

$$x^{2}-810x+7800-7800=-7800$$

Subtracting 7800 from itself leaves 0.

$$x^{2}-810x=-7800$$

Divide -810, the coefficient of the x term, by 2 to get -405. Then add the square of -405 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-810x+\left(-405\right)^{2}=-7800+\left(-405\right)^{2}$$

Square -405.

$$x^{2}-810x+164025=-7800+164025$$

Add -7800 to 164025.

$$x^{2}-810x+164025=156225$$

Factor x^{2}-810x+164025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-405\right)^{2}=156225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-405\right)^{2}}=\sqrt{156225}$$

Simplify.

$$x-405=5\sqrt{6249} x-405=-5\sqrt{6249}$$

Add 405 to both sides of the equation.

$$x=5\sqrt{6249}+405 x=405-5\sqrt{6249}$$