Solve 5000x^2-2870,52x-1698,31=0

$$5000 x ^ { 2 } - 2870,52 x - 1698,31 = 0$$

Solve for x

$$x=\frac{\sqrt{26378803169}+71763}{250000}\approx 0.936714105$$

$$x=\frac{71763-\sqrt{26378803169}}{250000}\approx -0.362610105$$

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.

$$5000x^{2}-2870,52x-1698,31=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5000 for a, -2870,52 for b, and -1698,31 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2870,52\right)±\sqrt{\left(-2870,52\right)^{2}-4\times 5000\left(-1698,31\right)}}{2\times 5000}$$

Square -2870,52 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-2870,52\right)±\sqrt{8239885,0704-4\times 5000\left(-1698,31\right)}}{2\times 5000}$$

Multiply -4 times 5000.

$$x=\frac{-\left(-2870,52\right)±\sqrt{8239885,0704-20000\left(-1698,31\right)}}{2\times 5000}$$

Multiply -20000 times -1698,31.

$$x=\frac{-\left(-2870,52\right)±\sqrt{8239885,0704+33966200}}{2\times 5000}$$

Add 8239885,0704 to 33966200.

$$x=\frac{-\left(-2870,52\right)±\sqrt{42206085,0704}}{2\times 5000}$$

Take the square root of 42206085,0704.

$$x=\frac{-\left(-2870,52\right)±\frac{\sqrt{26378803169}}{25}}{2\times 5000}$$

The opposite of -2870,52 is 2870,52.

$$x=\frac{2870,52±\frac{\sqrt{26378803169}}{25}}{2\times 5000}$$

Multiply 2 times 5000.

$$x=\frac{2870,52±\frac{\sqrt{26378803169}}{25}}{10000}$$

Now solve the equation x=\frac{2870,52±\frac{\sqrt{26378803169}}{25}}{10000} when ± is plus. Add 2870,52 to \frac{\sqrt{26378803169}}{25}.

$$x=\frac{\sqrt{26378803169}+71763}{25\times 10000}$$

Divide \frac{71763+\sqrt{26378803169}}{25} by 10000.

$$x=\frac{\sqrt{26378803169}+71763}{250000}$$

Now solve the equation x=\frac{2870,52±\frac{\sqrt{26378803169}}{25}}{10000} when ± is minus. Subtract \frac{\sqrt{26378803169}}{25} from 2870,52.

$$x=\frac{71763-\sqrt{26378803169}}{25\times 10000}$$

Divide \frac{71763-\sqrt{26378803169}}{25} by 10000.

$$x=\frac{71763-\sqrt{26378803169}}{250000}$$

The equation is now solved.

$$x=\frac{\sqrt{26378803169}+71763}{250000} x=\frac{71763-\sqrt{26378803169}}{250000}$$

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.

$$5000x^{2}-2870,52x-1698,31=0$$

Add 1698,31 to both sides of the equation.

$$5000x^{2}-2870,52x-1698,31-\left(-1698,31\right)=-\left(-1698,31\right)$$

Subtracting -1698,31 from itself leaves 0.

$$5000x^{2}-2870,52x=-\left(-1698,31\right)$$

Subtract -1698,31 from 0.

$$5000x^{2}-2870,52x=1698,31$$

Divide both sides by 5000.

$$\frac{5000x^{2}-2870,52x}{5000}=\frac{1698,31}{5000}$$

Dividing by 5000 undoes the multiplication by 5000.

$$x^{2}+\left(-\frac{2870,52}{5000}\right)x=\frac{1698,31}{5000}$$

Divide -2870,52 by 5000.

$$x^{2}-0,574104x=\frac{1698,31}{5000}$$

Divide 1698,31 by 5000.

$$x^{2}-0,574104x=0,339662$$

Divide -0,574104, the coefficient of the x term, by 2 to get -0,287052. Then add the square of -0,287052 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-0,574104x+\left(-0,287052\right)^{2}=0,339662+\left(-0,287052\right)^{2}$$

Square -0,287052 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-0,574104x+0,082398850704=0,339662+0,082398850704$$

Add 0,339662 to 0,082398850704 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-0,574104x+0,082398850704=0,422060850704$$

Factor x^{2}-0,574104x+0,082398850704. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0,287052\right)^{2}=0,422060850704$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0,287052\right)^{2}}=\sqrt{0,422060850704}$$

Simplify.

$$x-0,287052=\frac{\sqrt{26378803169}}{250000} x-0,287052=-\frac{\sqrt{26378803169}}{250000}$$

Add 0,287052 to both sides of the equation.

$$x=\frac{\sqrt{26378803169}+71763}{250000} x=\frac{71763-\sqrt{26378803169}}{250000}$$