Solve 1000x^2+612.5x+12.5=0

$$1000 { x }^{ 2 } +612.5x+12.5=0$$

Solve for x

$$x=\frac{\sqrt{2081}-49}{160}\approx -0.021137632$$

$$x=\frac{-\sqrt{2081}-49}{160}\approx -0.591362368$$

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.

$$1000x^{2}+612.5x+12.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1000 for a, 612.5 for b, and 12.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-612.5±\sqrt{612.5^{2}-4\times 1000\times 12.5}}{2\times 1000}$$

Square 612.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-612.5±\sqrt{375156.25-4\times 1000\times 12.5}}{2\times 1000}$$

Multiply -4 times 1000.

$$x=\frac{-612.5±\sqrt{375156.25-4000\times 12.5}}{2\times 1000}$$

Multiply -4000 times 12.5.

$$x=\frac{-612.5±\sqrt{375156.25-50000}}{2\times 1000}$$

Add 375156.25 to -50000.

$$x=\frac{-612.5±\sqrt{325156.25}}{2\times 1000}$$

Take the square root of 325156.25.

$$x=\frac{-612.5±\frac{25\sqrt{2081}}{2}}{2\times 1000}$$

Multiply 2 times 1000.

$$x=\frac{-612.5±\frac{25\sqrt{2081}}{2}}{2000}$$

Now solve the equation x=\frac{-612.5±\frac{25\sqrt{2081}}{2}}{2000} when ± is plus. Add -612.5 to \frac{25\sqrt{2081}}{2}.

$$x=\frac{25\sqrt{2081}-1225}{2\times 2000}$$

Divide \frac{-1225+25\sqrt{2081}}{2} by 2000.

$$x=\frac{\sqrt{2081}-49}{160}$$

Now solve the equation x=\frac{-612.5±\frac{25\sqrt{2081}}{2}}{2000} when ± is minus. Subtract \frac{25\sqrt{2081}}{2} from -612.5.

$$x=\frac{-25\sqrt{2081}-1225}{2\times 2000}$$

Divide \frac{-1225-25\sqrt{2081}}{2} by 2000.

$$x=\frac{-\sqrt{2081}-49}{160}$$

The equation is now solved.

$$x=\frac{\sqrt{2081}-49}{160} x=\frac{-\sqrt{2081}-49}{160}$$

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.

$$1000x^{2}+612.5x+12.5=0$$

Subtract 12.5 from both sides of the equation.

$$1000x^{2}+612.5x+12.5-12.5=-12.5$$

Subtracting 12.5 from itself leaves 0.

$$1000x^{2}+612.5x=-12.5$$

Divide both sides by 1000.

$$\frac{1000x^{2}+612.5x}{1000}=-\frac{12.5}{1000}$$

Dividing by 1000 undoes the multiplication by 1000.

$$x^{2}+\frac{612.5}{1000}x=-\frac{12.5}{1000}$$

Divide 612.5 by 1000.

$$x^{2}+0.6125x=-\frac{12.5}{1000}$$

Divide -12.5 by 1000.

$$x^{2}+0.6125x=-0.0125$$

Divide 0.6125, the coefficient of the x term, by 2 to get 0.30625. Then add the square of 0.30625 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+0.6125x+0.30625^{2}=-0.0125+0.30625^{2}$$

Square 0.30625 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+0.6125x+0.0937890625=-0.0125+0.0937890625$$

Add -0.0125 to 0.0937890625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+0.6125x+0.0937890625=0.0812890625$$

Factor x^{2}+0.6125x+0.0937890625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+0.30625\right)^{2}=0.0812890625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+0.30625\right)^{2}}=\sqrt{0.0812890625}$$

Simplify.

$$x+0.30625=\frac{\sqrt{2081}}{160} x+0.30625=-\frac{\sqrt{2081}}{160}$$

Subtract 0.30625 from both sides of the equation.

$$x=\frac{\sqrt{2081}-49}{160} x=\frac{-\sqrt{2081}-49}{160}$$