Solve -x^2+26x=1000000

$$- x ^ { 2 } + 26 x = 1000000$$

Solve for x (complex solution)

$$x=-\sqrt{999831}i+13\approx 13-999.91549643i$$

$$x=13+\sqrt{999831}i\approx 13+999.91549643i$$

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.

$$-x^{2}+26x=1000000$$

Subtract 1000000 from both sides of the equation.

$$-x^{2}+26x-1000000=1000000-1000000$$

Subtracting 1000000 from itself leaves 0.

$$-x^{2}+26x-1000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 26 for b, and -1000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-26±\sqrt{26^{2}-4\left(-1\right)\left(-1000000\right)}}{2\left(-1\right)}$$

Square 26.

$$x=\frac{-26±\sqrt{676-4\left(-1\right)\left(-1000000\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-26±\sqrt{676+4\left(-1000000\right)}}{2\left(-1\right)}$$

Multiply 4 times -1000000.

$$x=\frac{-26±\sqrt{676-4000000}}{2\left(-1\right)}$$

Add 676 to -4000000.

$$x=\frac{-26±\sqrt{-3999324}}{2\left(-1\right)}$$

Take the square root of -3999324.

$$x=\frac{-26±2\sqrt{999831}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-26±2\sqrt{999831}i}{-2}$$

Now solve the equation x=\frac{-26±2\sqrt{999831}i}{-2} when ± is plus. Add -26 to 2i\sqrt{999831}.

$$x=\frac{-26+2\sqrt{999831}i}{-2}$$

Divide -26+2i\sqrt{999831} by -2.

$$x=-\sqrt{999831}i+13$$

Now solve the equation x=\frac{-26±2\sqrt{999831}i}{-2} when ± is minus. Subtract 2i\sqrt{999831} from -26.

$$x=\frac{-2\sqrt{999831}i-26}{-2}$$

Divide -26-2i\sqrt{999831} by -2.

$$x=13+\sqrt{999831}i$$

The equation is now solved.

$$x=-\sqrt{999831}i+13 x=13+\sqrt{999831}i$$

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.

$$-x^{2}+26x=1000000$$

Divide both sides by -1.

$$\frac{-x^{2}+26x}{-1}=\frac{1000000}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{26}{-1}x=\frac{1000000}{-1}$$

Divide 26 by -1.

$$x^{2}-26x=\frac{1000000}{-1}$$

Divide 1000000 by -1.

$$x^{2}-26x=-1000000$$

Divide -26, the coefficient of the x term, by 2 to get -13. Then add the square of -13 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-26x+\left(-13\right)^{2}=-1000000+\left(-13\right)^{2}$$

Square -13.

$$x^{2}-26x+169=-1000000+169$$

Add -1000000 to 169.

$$x^{2}-26x+169=-999831$$

Factor x^{2}-26x+169. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-13\right)^{2}=-999831$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-13\right)^{2}}=\sqrt{-999831}$$

Simplify.

$$x-13=\sqrt{999831}i x-13=-\sqrt{999831}i$$

Add 13 to both sides of the equation.

$$x=13+\sqrt{999831}i x=-\sqrt{999831}i+13$$