Solve 52=-0.00211x^2+1.06x

$$52 = - 0.00211 x ^ { 2 } + 1.06 x$$

Solve for x

$$x = \frac{600 \sqrt{4755} + 53000}{211} \approx 447.269691087$$

$$x = \frac{53000 - 600 \sqrt{4755}}{211} \approx 55.09997716$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-0.00211x^{2}+1.06x=52$$

Subtract 52 from both sides.

$$-0.00211x^{2}+1.06x-52=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.00211 for a, 1.06 for b, and -52 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1.06±\sqrt{1.06^{2}-4\left(-0.00211\right)\left(-52\right)}}{2\left(-0.00211\right)}$$

Square 1.06 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-1.06±\sqrt{1.1236-4\left(-0.00211\right)\left(-52\right)}}{2\left(-0.00211\right)}$$

Multiply -4 times -0.00211.

$$x=\frac{-1.06±\sqrt{1.1236+0.00844\left(-52\right)}}{2\left(-0.00211\right)}$$

Multiply 0.00844 times -52.

$$x=\frac{-1.06±\sqrt{1.1236-0.43888}}{2\left(-0.00211\right)}$$

Add 1.1236 to -0.43888 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1.06±\sqrt{0.68472}}{2\left(-0.00211\right)}$$

Take the square root of 0.68472.

$$x=\frac{-1.06±\frac{3\sqrt{4755}}{250}}{2\left(-0.00211\right)}$$

Multiply 2 times -0.00211.

$$x=\frac{-1.06±\frac{3\sqrt{4755}}{250}}{-0.00422}$$

Now solve the equation x=\frac{-1.06±\frac{3\sqrt{4755}}{250}}{-0.00422} when ± is plus. Add -1.06 to \frac{3\sqrt{4755}}{250}.

$$x=\frac{\frac{3\sqrt{4755}}{250}-\frac{53}{50}}{-0.00422}$$

Divide -\frac{53}{50}+\frac{3\sqrt{4755}}{250} by -0.00422 by multiplying -\frac{53}{50}+\frac{3\sqrt{4755}}{250} by the reciprocal of -0.00422.

$$x=\frac{53000-600\sqrt{4755}}{211}$$

Now solve the equation x=\frac{-1.06±\frac{3\sqrt{4755}}{250}}{-0.00422} when ± is minus. Subtract \frac{3\sqrt{4755}}{250} from -1.06.

$$x=\frac{-\frac{3\sqrt{4755}}{250}-\frac{53}{50}}{-0.00422}$$

Divide -\frac{53}{50}-\frac{3\sqrt{4755}}{250} by -0.00422 by multiplying -\frac{53}{50}-\frac{3\sqrt{4755}}{250} by the reciprocal of -0.00422.

$$x=\frac{600\sqrt{4755}+53000}{211}$$

The equation is now solved.

$$x=\frac{53000-600\sqrt{4755}}{211} x=\frac{600\sqrt{4755}+53000}{211}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-0.00211x^{2}+1.06x=52$$

Divide both sides of the equation by -0.00211, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.00211x^{2}+1.06x}{-0.00211}=\frac{52}{-0.00211}$$

Dividing by -0.00211 undoes the multiplication by -0.00211.

$$x^{2}+\frac{1.06}{-0.00211}x=\frac{52}{-0.00211}$$

Divide 1.06 by -0.00211 by multiplying 1.06 by the reciprocal of -0.00211.

$$x^{2}-\frac{106000}{211}x=\frac{52}{-0.00211}$$

Divide 52 by -0.00211 by multiplying 52 by the reciprocal of -0.00211.

$$x^{2}-\frac{106000}{211}x=-\frac{5200000}{211}$$

Divide -\frac{106000}{211}, the coefficient of the x term, by 2 to get -\frac{53000}{211}. Then add the square of -\frac{53000}{211} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{106000}{211}x+\left(-\frac{53000}{211}\right)^{2}=-\frac{5200000}{211}+\left(-\frac{53000}{211}\right)^{2}$$

Square -\frac{53000}{211} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{106000}{211}x+\frac{2809000000}{44521}=-\frac{5200000}{211}+\frac{2809000000}{44521}$$

Add -\frac{5200000}{211} to \frac{2809000000}{44521} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{106000}{211}x+\frac{2809000000}{44521}=\frac{1711800000}{44521}$$

Factor x^{2}-\frac{106000}{211}x+\frac{2809000000}{44521}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{53000}{211}\right)^{2}=\frac{1711800000}{44521}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{53000}{211}\right)^{2}}=\sqrt{\frac{1711800000}{44521}}$$

Simplify.

$$x-\frac{53000}{211}=\frac{600\sqrt{4755}}{211} x-\frac{53000}{211}=-\frac{600\sqrt{4755}}{211}$$

Add \frac{53000}{211} to both sides of the equation.

$$x=\frac{600\sqrt{4755}+53000}{211} x=\frac{53000-600\sqrt{4755}}{211}$$