Solve 1300=85x-200-x^2

$$1300 = 85 x - 200 - x ^ { 2 }$$

Solve for x

$$x=25$$

$$x=60$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$85x-200-x^{2}=1300$$

Subtract 1300 from both sides.

$$85x-200-x^{2}-1300=0$$

Subtract 1300 from -200 to get -1500.

$$85x-1500-x^{2}=0$$

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.

$$-x^{2}+85x-1500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 85 for b, and -1500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-85±\sqrt{85^{2}-4\left(-1\right)\left(-1500\right)}}{2\left(-1\right)}$$

Square 85.

$$x=\frac{-85±\sqrt{7225-4\left(-1\right)\left(-1500\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-85±\sqrt{7225+4\left(-1500\right)}}{2\left(-1\right)}$$

Multiply 4 times -1500.

$$x=\frac{-85±\sqrt{7225-6000}}{2\left(-1\right)}$$

Add 7225 to -6000.

$$x=\frac{-85±\sqrt{1225}}{2\left(-1\right)}$$

Take the square root of 1225.

$$x=\frac{-85±35}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-85±35}{-2}$$

Now solve the equation x=\frac{-85±35}{-2} when ± is plus. Add -85 to 35.

$$x=-\frac{50}{-2}$$

Divide -50 by -2.

$$x=25$$

Now solve the equation x=\frac{-85±35}{-2} when ± is minus. Subtract 35 from -85.

$$x=-\frac{120}{-2}$$

Divide -120 by -2.

$$x=60$$

The equation is now solved.

$$x=25 x=60$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$85x-200-x^{2}=1300$$

Add 200 to both sides.

$$85x-x^{2}=1300+200$$

Add 1300 and 200 to get 1500.

$$85x-x^{2}=1500$$

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.

$$-x^{2}+85x=1500$$

Divide both sides by -1.

$$\frac{-x^{2}+85x}{-1}=\frac{1500}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{85}{-1}x=\frac{1500}{-1}$$

Divide 85 by -1.

$$x^{2}-85x=\frac{1500}{-1}$$

Divide 1500 by -1.

$$x^{2}-85x=-1500$$

Divide -85, the coefficient of the x term, by 2 to get -\frac{85}{2}. Then add the square of -\frac{85}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-85x+\left(-\frac{85}{2}\right)^{2}=-1500+\left(-\frac{85}{2}\right)^{2}$$

Square -\frac{85}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-85x+\frac{7225}{4}=-1500+\frac{7225}{4}$$

Add -1500 to \frac{7225}{4}.

$$x^{2}-85x+\frac{7225}{4}=\frac{1225}{4}$$

Factor x^{2}-85x+\frac{7225}{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}.

$$\left(x-\frac{85}{2}\right)^{2}=\frac{1225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{85}{2}\right)^{2}}=\sqrt{\frac{1225}{4}}$$

Simplify.

$$x-\frac{85}{2}=\frac{35}{2} x-\frac{85}{2}=-\frac{35}{2}$$

Add \frac{85}{2} to both sides of the equation.

$$x=60 x=25$$