Solve -37.46x^2+57768x+13100=0

$$-37.46 { x }^{ 2 } +57768x+13100 = 0$$

Solve for x

$$x = \frac{50 \sqrt{834776182} + 1444200}{1873} \approx 1542.351669071$$

$$x=\frac{1444200-50\sqrt{834776182}}{1873}\approx -0.226735809$$

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.

$$-37.46x^{2}+57768x+13100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -37.46 for a, 57768 for b, and 13100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-57768±\sqrt{57768^{2}-4\left(-37.46\right)\times 13100}}{2\left(-37.46\right)}$$

Square 57768.

$$x=\frac{-57768±\sqrt{3337141824-4\left(-37.46\right)\times 13100}}{2\left(-37.46\right)}$$

Multiply -4 times -37.46.

$$x=\frac{-57768±\sqrt{3337141824+149.84\times 13100}}{2\left(-37.46\right)}$$

Multiply 149.84 times 13100.

$$x=\frac{-57768±\sqrt{3337141824+1962904}}{2\left(-37.46\right)}$$

Add 3337141824 to 1962904.

$$x=\frac{-57768±\sqrt{3339104728}}{2\left(-37.46\right)}$$

Take the square root of 3339104728.

$$x=\frac{-57768±2\sqrt{834776182}}{2\left(-37.46\right)}$$

Multiply 2 times -37.46.

$$x=\frac{-57768±2\sqrt{834776182}}{-74.92}$$

Now solve the equation x=\frac{-57768±2\sqrt{834776182}}{-74.92} when ± is plus. Add -57768 to 2\sqrt{834776182}.

$$x=\frac{2\sqrt{834776182}-57768}{-74.92}$$

Divide -57768+2\sqrt{834776182} by -74.92 by multiplying -57768+2\sqrt{834776182} by the reciprocal of -74.92.

$$x=\frac{1444200-50\sqrt{834776182}}{1873}$$

Now solve the equation x=\frac{-57768±2\sqrt{834776182}}{-74.92} when ± is minus. Subtract 2\sqrt{834776182} from -57768.

$$x=\frac{-2\sqrt{834776182}-57768}{-74.92}$$

Divide -57768-2\sqrt{834776182} by -74.92 by multiplying -57768-2\sqrt{834776182} by the reciprocal of -74.92.

$$x=\frac{50\sqrt{834776182}+1444200}{1873}$$

The equation is now solved.

$$x=\frac{1444200-50\sqrt{834776182}}{1873} x=\frac{50\sqrt{834776182}+1444200}{1873}$$

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.

$$-37.46x^{2}+57768x+13100=0$$

Subtract 13100 from both sides of the equation.

$$-37.46x^{2}+57768x+13100-13100=-13100$$

Subtracting 13100 from itself leaves 0.

$$-37.46x^{2}+57768x=-13100$$

Divide both sides of the equation by -37.46, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-37.46x^{2}+57768x}{-37.46}=-\frac{13100}{-37.46}$$

Dividing by -37.46 undoes the multiplication by -37.46.

$$x^{2}+\frac{57768}{-37.46}x=-\frac{13100}{-37.46}$$

Divide 57768 by -37.46 by multiplying 57768 by the reciprocal of -37.46.

$$x^{2}-\frac{2888400}{1873}x=-\frac{13100}{-37.46}$$

Divide -13100 by -37.46 by multiplying -13100 by the reciprocal of -37.46.

$$x^{2}-\frac{2888400}{1873}x=\frac{655000}{1873}$$

Divide -\frac{2888400}{1873}, the coefficient of the x term, by 2 to get -\frac{1444200}{1873}. Then add the square of -\frac{1444200}{1873} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2888400}{1873}x+\left(-\frac{1444200}{1873}\right)^{2}=\frac{655000}{1873}+\left(-\frac{1444200}{1873}\right)^{2}$$

Square -\frac{1444200}{1873} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2888400}{1873}x+\frac{2085713640000}{3508129}=\frac{655000}{1873}+\frac{2085713640000}{3508129}$$

Add \frac{655000}{1873} to \frac{2085713640000}{3508129} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2888400}{1873}x+\frac{2085713640000}{3508129}=\frac{2086940455000}{3508129}$$

Factor x^{2}-\frac{2888400}{1873}x+\frac{2085713640000}{3508129}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1444200}{1873}\right)^{2}=\frac{2086940455000}{3508129}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1444200}{1873}\right)^{2}}=\sqrt{\frac{2086940455000}{3508129}}$$

Simplify.

$$x-\frac{1444200}{1873}=\frac{50\sqrt{834776182}}{1873} x-\frac{1444200}{1873}=-\frac{50\sqrt{834776182}}{1873}$$

Add \frac{1444200}{1873} to both sides of the equation.

$$x=\frac{50\sqrt{834776182}+1444200}{1873} x=\frac{1444200-50\sqrt{834776182}}{1873}$$