Solve -x^2+52x+640=540

$$- { x }^{ 2 } +52x+640=540$$

Solve for x

$$x=2\sqrt{194}+26\approx 53.856776554$$

$$x=26-2\sqrt{194}\approx -1.856776554$$

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.

$$-x^{2}+52x+640=540$$

Subtract 540 from both sides of the equation.

$$-x^{2}+52x+640-540=540-540$$

Subtracting 540 from itself leaves 0.

$$-x^{2}+52x+640-540=0$$

Subtract 540 from 640.

$$-x^{2}+52x+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 52 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-52±\sqrt{52^{2}-4\left(-1\right)\times 100}}{2\left(-1\right)}$$

Square 52.

$$x=\frac{-52±\sqrt{2704-4\left(-1\right)\times 100}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-52±\sqrt{2704+4\times 100}}{2\left(-1\right)}$$

Multiply 4 times 100.

$$x=\frac{-52±\sqrt{2704+400}}{2\left(-1\right)}$$

Add 2704 to 400.

$$x=\frac{-52±\sqrt{3104}}{2\left(-1\right)}$$

Take the square root of 3104.

$$x=\frac{-52±4\sqrt{194}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-52±4\sqrt{194}}{-2}$$

Now solve the equation x=\frac{-52±4\sqrt{194}}{-2} when ± is plus. Add -52 to 4\sqrt{194}.

$$x=\frac{4\sqrt{194}-52}{-2}$$

Divide -52+4\sqrt{194} by -2.

$$x=26-2\sqrt{194}$$

Now solve the equation x=\frac{-52±4\sqrt{194}}{-2} when ± is minus. Subtract 4\sqrt{194} from -52.

$$x=\frac{-4\sqrt{194}-52}{-2}$$

Divide -52-4\sqrt{194} by -2.

$$x=2\sqrt{194}+26$$

The equation is now solved.

$$x=26-2\sqrt{194} x=2\sqrt{194}+26$$

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.

$$-x^{2}+52x+640=540$$

Subtract 640 from both sides of the equation.

$$-x^{2}+52x+640-640=540-640$$

Subtracting 640 from itself leaves 0.

$$-x^{2}+52x=540-640$$

Subtract 640 from 540.

$$-x^{2}+52x=-100$$

Divide both sides by -1.

$$\frac{-x^{2}+52x}{-1}=-\frac{100}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{52}{-1}x=-\frac{100}{-1}$$

Divide 52 by -1.

$$x^{2}-52x=-\frac{100}{-1}$$

Divide -100 by -1.

$$x^{2}-52x=100$$

Divide -52, the coefficient of the x term, by 2 to get -26. Then add the square of -26 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-52x+\left(-26\right)^{2}=100+\left(-26\right)^{2}$$

Square -26.

$$x^{2}-52x+676=100+676$$

Add 100 to 676.

$$x^{2}-52x+676=776$$

Factor x^{2}-52x+676. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-26\right)^{2}=776$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-26\right)^{2}}=\sqrt{776}$$

Simplify.

$$x-26=2\sqrt{194} x-26=-2\sqrt{194}$$

Add 26 to both sides of the equation.

$$x=2\sqrt{194}+26 x=26-2\sqrt{194}$$