Solve 2033.88-5800.56x+3681.2x^2=0

$$2033.88-5800.56x+3681.2 { x }^{ 2 } =0$$

Solve for x

$$x = \frac{\sqrt{577815639} + 72507}{92030} \approx 1.049057873$$

$$x=\frac{72507-\sqrt{577815639}}{92030}\approx 0.526667434$$

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.

$$3681.2x^{2}-5800.56x+2033.88=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3681.2 for a, -5800.56 for b, and 2033.88 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5800.56\right)±\sqrt{\left(-5800.56\right)^{2}-4\times 3681.2\times 2033.88}}{2\times 3681.2}$$

Square -5800.56 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-5800.56\right)±\sqrt{33646496.3136-4\times 3681.2\times 2033.88}}{2\times 3681.2}$$

Multiply -4 times 3681.2.

$$x=\frac{-\left(-5800.56\right)±\sqrt{33646496.3136-14724.8\times 2033.88}}{2\times 3681.2}$$

Multiply -14724.8 times 2033.88 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-5800.56\right)±\sqrt{33646496.3136-29948476.224}}{2\times 3681.2}$$

Add 33646496.3136 to -29948476.224 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-5800.56\right)±\sqrt{3698020.0896}}{2\times 3681.2}$$

Take the square root of 3698020.0896.

$$x=\frac{-\left(-5800.56\right)±\frac{2\sqrt{577815639}}{25}}{2\times 3681.2}$$

The opposite of -5800.56 is 5800.56.

$$x=\frac{5800.56±\frac{2\sqrt{577815639}}{25}}{2\times 3681.2}$$

Multiply 2 times 3681.2.

$$x=\frac{5800.56±\frac{2\sqrt{577815639}}{25}}{7362.4}$$

Now solve the equation x=\frac{5800.56±\frac{2\sqrt{577815639}}{25}}{7362.4} when ± is plus. Add 5800.56 to \frac{2\sqrt{577815639}}{25}.

$$x=\frac{2\sqrt{577815639}+145014}{25\times 7362.4}$$

Divide \frac{145014+2\sqrt{577815639}}{25} by 7362.4 by multiplying \frac{145014+2\sqrt{577815639}}{25} by the reciprocal of 7362.4.

$$x=\frac{\sqrt{577815639}+72507}{92030}$$

Now solve the equation x=\frac{5800.56±\frac{2\sqrt{577815639}}{25}}{7362.4} when ± is minus. Subtract \frac{2\sqrt{577815639}}{25} from 5800.56.

$$x=\frac{145014-2\sqrt{577815639}}{25\times 7362.4}$$

Divide \frac{145014-2\sqrt{577815639}}{25} by 7362.4 by multiplying \frac{145014-2\sqrt{577815639}}{25} by the reciprocal of 7362.4.

$$x=\frac{72507-\sqrt{577815639}}{92030}$$

The equation is now solved.

$$x=\frac{\sqrt{577815639}+72507}{92030} x=\frac{72507-\sqrt{577815639}}{92030}$$

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.

$$3681.2x^{2}-5800.56x+2033.88=0$$

Subtract 2033.88 from both sides of the equation.

$$3681.2x^{2}-5800.56x+2033.88-2033.88=-2033.88$$

Subtracting 2033.88 from itself leaves 0.

$$3681.2x^{2}-5800.56x=-2033.88$$

Divide both sides of the equation by 3681.2, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{3681.2x^{2}-5800.56x}{3681.2}=-\frac{2033.88}{3681.2}$$

Dividing by 3681.2 undoes the multiplication by 3681.2.

$$x^{2}+\left(-\frac{5800.56}{3681.2}\right)x=-\frac{2033.88}{3681.2}$$

Divide -5800.56 by 3681.2 by multiplying -5800.56 by the reciprocal of 3681.2.

$$x^{2}-\frac{72507}{46015}x=-\frac{2033.88}{3681.2}$$

Divide -2033.88 by 3681.2 by multiplying -2033.88 by the reciprocal of 3681.2.

$$x^{2}-\frac{72507}{46015}x=-\frac{50847}{92030}$$

Divide -\frac{72507}{46015}, the coefficient of the x term, by 2 to get -\frac{72507}{92030}. Then add the square of -\frac{72507}{92030} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{72507}{46015}x+\left(-\frac{72507}{92030}\right)^{2}=-\frac{50847}{92030}+\left(-\frac{72507}{92030}\right)^{2}$$

Square -\frac{72507}{92030} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{72507}{46015}x+\frac{5257265049}{8469520900}=-\frac{50847}{92030}+\frac{5257265049}{8469520900}$$

Add -\frac{50847}{92030} to \frac{5257265049}{8469520900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{72507}{46015}x+\frac{5257265049}{8469520900}=\frac{577815639}{8469520900}$$

Factor x^{2}-\frac{72507}{46015}x+\frac{5257265049}{8469520900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{72507}{92030}\right)^{2}=\frac{577815639}{8469520900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{72507}{92030}\right)^{2}}=\sqrt{\frac{577815639}{8469520900}}$$

Simplify.

$$x-\frac{72507}{92030}=\frac{\sqrt{577815639}}{92030} x-\frac{72507}{92030}=-\frac{\sqrt{577815639}}{92030}$$

Add \frac{72507}{92030} to both sides of the equation.

$$x=\frac{\sqrt{577815639}+72507}{92030} x=\frac{72507-\sqrt{577815639}}{92030}$$