Solve 1000x^2+999x+77=6

$$1000 x ^ { 2 } + 999 x + 77 = 6$$

Solve for x

$$x=\frac{\sqrt{714001}-999}{2000}\approx -0.077007101$$

$$x=\frac{-\sqrt{714001}-999}{2000}\approx -0.921992899$$

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.

$$1000x^{2}+999x+77=6$$

Subtract 6 from both sides of the equation.

$$1000x^{2}+999x+77-6=6-6$$

Subtracting 6 from itself leaves 0.

$$1000x^{2}+999x+77-6=0$$

Subtract 6 from 77.

$$1000x^{2}+999x+71=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1000 for a, 999 for b, and 71 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-999±\sqrt{999^{2}-4\times 1000\times 71}}{2\times 1000}$$

Square 999.

$$x=\frac{-999±\sqrt{998001-4\times 1000\times 71}}{2\times 1000}$$

Multiply -4 times 1000.

$$x=\frac{-999±\sqrt{998001-4000\times 71}}{2\times 1000}$$

Multiply -4000 times 71.

$$x=\frac{-999±\sqrt{998001-284000}}{2\times 1000}$$

Add 998001 to -284000.

$$x=\frac{-999±\sqrt{714001}}{2\times 1000}$$

Multiply 2 times 1000.

$$x=\frac{-999±\sqrt{714001}}{2000}$$

Now solve the equation x=\frac{-999±\sqrt{714001}}{2000} when ± is plus. Add -999 to \sqrt{714001}.

$$x=\frac{\sqrt{714001}-999}{2000}$$

Now solve the equation x=\frac{-999±\sqrt{714001}}{2000} when ± is minus. Subtract \sqrt{714001} from -999.

$$x=\frac{-\sqrt{714001}-999}{2000}$$

The equation is now solved.

$$x=\frac{\sqrt{714001}-999}{2000} x=\frac{-\sqrt{714001}-999}{2000}$$

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.

$$1000x^{2}+999x+77=6$$

Subtract 77 from both sides of the equation.

$$1000x^{2}+999x+77-77=6-77$$

Subtracting 77 from itself leaves 0.

$$1000x^{2}+999x=6-77$$

Subtract 77 from 6.

$$1000x^{2}+999x=-71$$

Divide both sides by 1000.

$$\frac{1000x^{2}+999x}{1000}=-\frac{71}{1000}$$

Dividing by 1000 undoes the multiplication by 1000.

$$x^{2}+\frac{999}{1000}x=-\frac{71}{1000}$$

Divide \frac{999}{1000}, the coefficient of the x term, by 2 to get \frac{999}{2000}. Then add the square of \frac{999}{2000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{999}{1000}x+\left(\frac{999}{2000}\right)^{2}=-\frac{71}{1000}+\left(\frac{999}{2000}\right)^{2}$$

Square \frac{999}{2000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{999}{1000}x+\frac{998001}{4000000}=-\frac{71}{1000}+\frac{998001}{4000000}$$

Add -\frac{71}{1000} to \frac{998001}{4000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{999}{1000}x+\frac{998001}{4000000}=\frac{714001}{4000000}$$

Factor x^{2}+\frac{999}{1000}x+\frac{998001}{4000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{999}{2000}\right)^{2}=\frac{714001}{4000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{999}{2000}\right)^{2}}=\sqrt{\frac{714001}{4000000}}$$

Simplify.

$$x+\frac{999}{2000}=\frac{\sqrt{714001}}{2000} x+\frac{999}{2000}=-\frac{\sqrt{714001}}{2000}$$

Subtract \frac{999}{2000} from both sides of the equation.

$$x=\frac{\sqrt{714001}-999}{2000} x=\frac{-\sqrt{714001}-999}{2000}$$