Solve -x^2+2019.5x=x-2019.5

$$- { x }^{ 2 } +2019.5x = x-2019.5$$

Solve for x

$$x=-1$$

$$x=2019.5$$

Steps Using the Quadratic Formula

Subtract x from both sides.

$$-x^{2}+2019.5x-x=-2019.5$$

Combine 2019.5x and -x to get 2018.5x.

$$-x^{2}+2018.5x=-2019.5$$

Add 2019.5 to both sides.

$$-x^{2}+2018.5x+2019.5=0$$

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.

$$-x^{2}+2018.5x+\frac{4039}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2018.5 for b, and \frac{4039}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2018.5±\sqrt{2018.5^{2}-4\left(-1\right)\times \frac{4039}{2}}}{2\left(-1\right)}$$

Square 2018.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-2018.5±\sqrt{4074342.25-4\left(-1\right)\times \frac{4039}{2}}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-2018.5±\sqrt{4074342.25+4\times \frac{4039}{2}}}{2\left(-1\right)}$$

Multiply 4 times \frac{4039}{2}.

$$x=\frac{-2018.5±\sqrt{4074342.25+8078}}{2\left(-1\right)}$$

Add 4074342.25 to 8078.

$$x=\frac{-2018.5±\sqrt{4082420.25}}{2\left(-1\right)}$$

Take the square root of 4082420.25.

$$x=\frac{-2018.5±\frac{4041}{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2018.5±\frac{4041}{2}}{-2}$$

Now solve the equation x=\frac{-2018.5±\frac{4041}{2}}{-2} when ± is plus. Add -2018.5 to \frac{4041}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide 2 by -2.

$$x=-1$$

Now solve the equation x=\frac{-2018.5±\frac{4041}{2}}{-2} when ± is minus. Subtract \frac{4041}{2} from -2018.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{4039}{-2}$$

Divide -4039 by -2.

$$x=\frac{4039}{2}$$

The equation is now solved.

$$x=-1 x=\frac{4039}{2}$$

Steps for Completing the Square

Subtract x from both sides.

$$-x^{2}+2019.5x-x=-2019.5$$

Combine 2019.5x and -x to get 2018.5x.

$$-x^{2}+2018.5x=-2019.5$$

Divide both sides by -1.

$$\frac{-x^{2}+2018.5x}{-1}=-\frac{2019.5}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{2018.5}{-1}x=-\frac{2019.5}{-1}$$

Divide 2018.5 by -1.

$$x^{2}-2018.5x=-\frac{2019.5}{-1}$$

Divide -2019.5 by -1.

$$x^{2}-2018.5x=2019.5$$

Divide -2018.5, the coefficient of the x term, by 2 to get -1009.25. Then add the square of -1009.25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2018.5x+\left(-1009.25\right)^{2}=2019.5+\left(-1009.25\right)^{2}$$

Square -1009.25 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-2018.5x+1018585.5625=2019.5+1018585.5625$$

Add 2019.5 to 1018585.5625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-2018.5x+1018585.5625=1020605.0625$$

Factor x^{2}-2018.5x+1018585.5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1009.25\right)^{2}=1020605.0625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1009.25\right)^{2}}=\sqrt{1020605.0625}$$

Simplify.

$$x-1009.25=\frac{4041}{4} x-1009.25=-\frac{4041}{4}$$

Add 1009.25 to both sides of the equation.

$$x=\frac{4039}{2} x=-1$$