Solve -37.46x^2+57768x+13.1*1000000=0

$$-37.46 { x }^{ 2 } +57768x+13.1 \times 1000000 = 0$$

Solve for x

$$x = \frac{800 \sqrt{5175826} + 1444200}{1873} \approx 1742.784484953$$

$$x=\frac{1444200-800\sqrt{5175826}}{1873}\approx -200.65955169$$

Steps Using the Quadratic Formula

Multiply 13.1 and 1000000 to get 13100000.

$$-37.46x^{2}+57768x+13100000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -37.46 for a, 57768 for b, and 13100000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-57768±\sqrt{57768^{2}-4\left(-37.46\right)\times 13100000}}{2\left(-37.46\right)}$$

Square 57768.

$$x=\frac{-57768±\sqrt{3337141824-4\left(-37.46\right)\times 13100000}}{2\left(-37.46\right)}$$

Multiply -4 times -37.46.

$$x=\frac{-57768±\sqrt{3337141824+149.84\times 13100000}}{2\left(-37.46\right)}$$

Multiply 149.84 times 13100000.

$$x=\frac{-57768±\sqrt{3337141824+1962904000}}{2\left(-37.46\right)}$$

Add 3337141824 to 1962904000.

$$x=\frac{-57768±\sqrt{5300045824}}{2\left(-37.46\right)}$$

Take the square root of 5300045824.

$$x=\frac{-57768±32\sqrt{5175826}}{2\left(-37.46\right)}$$

Multiply 2 times -37.46.

$$x=\frac{-57768±32\sqrt{5175826}}{-74.92}$$

Now solve the equation x=\frac{-57768±32\sqrt{5175826}}{-74.92} when ± is plus. Add -57768 to 32\sqrt{5175826}.

$$x=\frac{32\sqrt{5175826}-57768}{-74.92}$$

Divide -57768+32\sqrt{5175826} by -74.92 by multiplying -57768+32\sqrt{5175826} by the reciprocal of -74.92.

$$x=\frac{1444200-800\sqrt{5175826}}{1873}$$

Now solve the equation x=\frac{-57768±32\sqrt{5175826}}{-74.92} when ± is minus. Subtract 32\sqrt{5175826} from -57768.

$$x=\frac{-32\sqrt{5175826}-57768}{-74.92}$$

Divide -57768-32\sqrt{5175826} by -74.92 by multiplying -57768-32\sqrt{5175826} by the reciprocal of -74.92.

$$x=\frac{800\sqrt{5175826}+1444200}{1873}$$

The equation is now solved.

$$x=\frac{1444200-800\sqrt{5175826}}{1873} x=\frac{800\sqrt{5175826}+1444200}{1873}$$

Steps for Completing the Square

Multiply 13.1 and 1000000 to get 13100000.

$$-37.46x^{2}+57768x+13100000=0$$

Subtract 13100000 from both sides. Anything subtracted from zero gives its negation.

$$-37.46x^{2}+57768x=-13100000$$

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

$$\frac{-37.46x^{2}+57768x}{-37.46}=-\frac{13100000}{-37.46}$$

Dividing by -37.46 undoes the multiplication by -37.46.

$$x^{2}+\frac{57768}{-37.46}x=-\frac{13100000}{-37.46}$$

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

$$x^{2}-\frac{2888400}{1873}x=-\frac{13100000}{-37.46}$$

Divide -13100000 by -37.46 by multiplying -13100000 by the reciprocal of -37.46.

$$x^{2}-\frac{2888400}{1873}x=\frac{655000000}{1873}$$

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.

$$x^{2}-\frac{2888400}{1873}x+\left(-\frac{1444200}{1873}\right)^{2}=\frac{655000000}{1873}+\left(-\frac{1444200}{1873}\right)^{2}$$

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

$$x^{2}-\frac{2888400}{1873}x+\frac{2085713640000}{3508129}=\frac{655000000}{1873}+\frac{2085713640000}{3508129}$$

Add \frac{655000000}{1873} to \frac{2085713640000}{3508129} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2888400}{1873}x+\frac{2085713640000}{3508129}=\frac{3312528640000}{3508129}$$

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}.

$$\left(x-\frac{1444200}{1873}\right)^{2}=\frac{3312528640000}{3508129}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1444200}{1873}\right)^{2}}=\sqrt{\frac{3312528640000}{3508129}}$$

Simplify.

$$x-\frac{1444200}{1873}=\frac{800\sqrt{5175826}}{1873} x-\frac{1444200}{1873}=-\frac{800\sqrt{5175826}}{1873}$$

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

$$x=\frac{800\sqrt{5175826}+1444200}{1873} x=\frac{1444200-800\sqrt{5175826}}{1873}$$