Solve 1000x^2+6125x+125=0

$$1000 { x }^{ 2 } +6125x+125=0$$

Solve for x

$$x=\frac{\sqrt{2369}-49}{16}\approx -0.020476619$$

$$x=\frac{-\sqrt{2369}-49}{16}\approx -6.104523381$$

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.

$$1000x^{2}+6125x+125=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1000 for a, 6125 for b, and 125 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6125±\sqrt{6125^{2}-4\times 1000\times 125}}{2\times 1000}$$

Square 6125.

$$x=\frac{-6125±\sqrt{37515625-4\times 1000\times 125}}{2\times 1000}$$

Multiply -4 times 1000.

$$x=\frac{-6125±\sqrt{37515625-4000\times 125}}{2\times 1000}$$

Multiply -4000 times 125.

$$x=\frac{-6125±\sqrt{37515625-500000}}{2\times 1000}$$

Add 37515625 to -500000.

$$x=\frac{-6125±\sqrt{37015625}}{2\times 1000}$$

Take the square root of 37015625.

$$x=\frac{-6125±125\sqrt{2369}}{2\times 1000}$$

Multiply 2 times 1000.

$$x=\frac{-6125±125\sqrt{2369}}{2000}$$

Now solve the equation x=\frac{-6125±125\sqrt{2369}}{2000} when ± is plus. Add -6125 to 125\sqrt{2369}.

$$x=\frac{125\sqrt{2369}-6125}{2000}$$

Divide -6125+125\sqrt{2369} by 2000.

$$x=\frac{\sqrt{2369}-49}{16}$$

Now solve the equation x=\frac{-6125±125\sqrt{2369}}{2000} when ± is minus. Subtract 125\sqrt{2369} from -6125.

$$x=\frac{-125\sqrt{2369}-6125}{2000}$$

Divide -6125-125\sqrt{2369} by 2000.

$$x=\frac{-\sqrt{2369}-49}{16}$$

The equation is now solved.

$$x=\frac{\sqrt{2369}-49}{16} x=\frac{-\sqrt{2369}-49}{16}$$

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.

$$1000x^{2}+6125x+125=0$$

Subtract 125 from both sides of the equation.

$$1000x^{2}+6125x+125-125=-125$$

Subtracting 125 from itself leaves 0.

$$1000x^{2}+6125x=-125$$

Divide both sides by 1000.

$$\frac{1000x^{2}+6125x}{1000}=-\frac{125}{1000}$$

Dividing by 1000 undoes the multiplication by 1000.

$$x^{2}+\frac{6125}{1000}x=-\frac{125}{1000}$$

Reduce the fraction \frac{6125}{1000} to lowest terms by extracting and canceling out 125.

$$x^{2}+\frac{49}{8}x=-\frac{125}{1000}$$

Reduce the fraction \frac{-125}{1000} to lowest terms by extracting and canceling out 125.

$$x^{2}+\frac{49}{8}x=-\frac{1}{8}$$

Divide \frac{49}{8}, the coefficient of the x term, by 2 to get \frac{49}{16}. Then add the square of \frac{49}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{49}{8}x+\left(\frac{49}{16}\right)^{2}=-\frac{1}{8}+\left(\frac{49}{16}\right)^{2}$$

Square \frac{49}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{49}{8}x+\frac{2401}{256}=-\frac{1}{8}+\frac{2401}{256}$$

Add -\frac{1}{8} to \frac{2401}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{49}{8}x+\frac{2401}{256}=\frac{2369}{256}$$

Factor x^{2}+\frac{49}{8}x+\frac{2401}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{49}{16}\right)^{2}=\frac{2369}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{49}{16}\right)^{2}}=\sqrt{\frac{2369}{256}}$$

Simplify.

$$x+\frac{49}{16}=\frac{\sqrt{2369}}{16} x+\frac{49}{16}=-\frac{\sqrt{2369}}{16}$$

Subtract \frac{49}{16} from both sides of the equation.

$$x=\frac{\sqrt{2369}-49}{16} x=\frac{-\sqrt{2369}-49}{16}$$