Solve -1000x^2+17000x+30000=0

$$-1000 { x }^{ 2 } +17000x+30000=0$$

Solve for x

$$x = \frac{\sqrt{409} + 17}{2} \approx 18.611874208$$

$$x=\frac{17-\sqrt{409}}{2}\approx -1.611874208$$

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.

$$-1000x^{2}+17000x+30000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1000 for a, 17000 for b, and 30000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17000±\sqrt{17000^{2}-4\left(-1000\right)\times 30000}}{2\left(-1000\right)}$$

Square 17000.

$$x=\frac{-17000±\sqrt{289000000-4\left(-1000\right)\times 30000}}{2\left(-1000\right)}$$

Multiply -4 times -1000.

$$x=\frac{-17000±\sqrt{289000000+4000\times 30000}}{2\left(-1000\right)}$$

Multiply 4000 times 30000.

$$x=\frac{-17000±\sqrt{289000000+120000000}}{2\left(-1000\right)}$$

Add 289000000 to 120000000.

$$x=\frac{-17000±\sqrt{409000000}}{2\left(-1000\right)}$$

Take the square root of 409000000.

$$x=\frac{-17000±1000\sqrt{409}}{2\left(-1000\right)}$$

Multiply 2 times -1000.

$$x=\frac{-17000±1000\sqrt{409}}{-2000}$$

Now solve the equation x=\frac{-17000±1000\sqrt{409}}{-2000} when ± is plus. Add -17000 to 1000\sqrt{409}.

$$x=\frac{1000\sqrt{409}-17000}{-2000}$$

Divide -17000+1000\sqrt{409} by -2000.

$$x=\frac{17-\sqrt{409}}{2}$$

Now solve the equation x=\frac{-17000±1000\sqrt{409}}{-2000} when ± is minus. Subtract 1000\sqrt{409} from -17000.

$$x=\frac{-1000\sqrt{409}-17000}{-2000}$$

Divide -17000-1000\sqrt{409} by -2000.

$$x=\frac{\sqrt{409}+17}{2}$$

The equation is now solved.

$$x=\frac{17-\sqrt{409}}{2} x=\frac{\sqrt{409}+17}{2}$$

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.

$$-1000x^{2}+17000x+30000=0$$

Subtract 30000 from both sides of the equation.

$$-1000x^{2}+17000x+30000-30000=-30000$$

Subtracting 30000 from itself leaves 0.

$$-1000x^{2}+17000x=-30000$$

Divide both sides by -1000.

$$\frac{-1000x^{2}+17000x}{-1000}=-\frac{30000}{-1000}$$

Dividing by -1000 undoes the multiplication by -1000.

$$x^{2}+\frac{17000}{-1000}x=-\frac{30000}{-1000}$$

Divide 17000 by -1000.

$$x^{2}-17x=-\frac{30000}{-1000}$$

Divide -30000 by -1000.

$$x^{2}-17x=30$$

Divide -17, the coefficient of the x term, by 2 to get -\frac{17}{2}. Then add the square of -\frac{17}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=30+\left(-\frac{17}{2}\right)^{2}$$

Square -\frac{17}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-17x+\frac{289}{4}=30+\frac{289}{4}$$

Add 30 to \frac{289}{4}.

$$x^{2}-17x+\frac{289}{4}=\frac{409}{4}$$

Factor x^{2}-17x+\frac{289}{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}.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{409}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{409}{4}}$$

Simplify.

$$x-\frac{17}{2}=\frac{\sqrt{409}}{2} x-\frac{17}{2}=-\frac{\sqrt{409}}{2}$$

Add \frac{17}{2} to both sides of the equation.

$$x=\frac{\sqrt{409}+17}{2} x=\frac{17-\sqrt{409}}{2}$$