Solve -110=37587x+-491x^2

$$-110 = 37587x+-491 { x }^{ 2 }$$

Solve for x

$$x = \frac{\sqrt{1412998609} + 37587}{982} \approx 76.554861259$$

$$x=\frac{37587-\sqrt{1412998609}}{982}\approx -0.002926432$$

Steps Using the Quadratic Formula

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

$$37587x-491x^{2}=-110$$

Add 110 to both sides.

$$37587x-491x^{2}+110=0$$

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.

$$-491x^{2}+37587x+110=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -491 for a, 37587 for b, and 110 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-37587±\sqrt{37587^{2}-4\left(-491\right)\times 110}}{2\left(-491\right)}$$

Square 37587.

$$x=\frac{-37587±\sqrt{1412782569-4\left(-491\right)\times 110}}{2\left(-491\right)}$$

Multiply -4 times -491.

$$x=\frac{-37587±\sqrt{1412782569+1964\times 110}}{2\left(-491\right)}$$

Multiply 1964 times 110.

$$x=\frac{-37587±\sqrt{1412782569+216040}}{2\left(-491\right)}$$

Add 1412782569 to 216040.

$$x=\frac{-37587±\sqrt{1412998609}}{2\left(-491\right)}$$

Multiply 2 times -491.

$$x=\frac{-37587±\sqrt{1412998609}}{-982}$$

Now solve the equation x=\frac{-37587±\sqrt{1412998609}}{-982} when ± is plus. Add -37587 to \sqrt{1412998609}.

$$x=\frac{\sqrt{1412998609}-37587}{-982}$$

Divide -37587+\sqrt{1412998609} by -982.

$$x=\frac{37587-\sqrt{1412998609}}{982}$$

Now solve the equation x=\frac{-37587±\sqrt{1412998609}}{-982} when ± is minus. Subtract \sqrt{1412998609} from -37587.

$$x=\frac{-\sqrt{1412998609}-37587}{-982}$$

Divide -37587-\sqrt{1412998609} by -982.

$$x=\frac{\sqrt{1412998609}+37587}{982}$$

The equation is now solved.

$$x=\frac{37587-\sqrt{1412998609}}{982} x=\frac{\sqrt{1412998609}+37587}{982}$$

Steps for Completing the Square

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

$$37587x-491x^{2}=-110$$

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.

$$-491x^{2}+37587x=-110$$

Divide both sides by -491.

$$\frac{-491x^{2}+37587x}{-491}=-\frac{110}{-491}$$

Dividing by -491 undoes the multiplication by -491.

$$x^{2}+\frac{37587}{-491}x=-\frac{110}{-491}$$

Divide 37587 by -491.

$$x^{2}-\frac{37587}{491}x=-\frac{110}{-491}$$

Divide -110 by -491.

$$x^{2}-\frac{37587}{491}x=\frac{110}{491}$$

Divide -\frac{37587}{491}, the coefficient of the x term, by 2 to get -\frac{37587}{982}. Then add the square of -\frac{37587}{982} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{37587}{491}x+\left(-\frac{37587}{982}\right)^{2}=\frac{110}{491}+\left(-\frac{37587}{982}\right)^{2}$$

Square -\frac{37587}{982} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{37587}{491}x+\frac{1412782569}{964324}=\frac{110}{491}+\frac{1412782569}{964324}$$

Add \frac{110}{491} to \frac{1412782569}{964324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{37587}{491}x+\frac{1412782569}{964324}=\frac{1412998609}{964324}$$

Factor x^{2}-\frac{37587}{491}x+\frac{1412782569}{964324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{37587}{982}\right)^{2}=\frac{1412998609}{964324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{37587}{982}\right)^{2}}=\sqrt{\frac{1412998609}{964324}}$$

Simplify.

$$x-\frac{37587}{982}=\frac{\sqrt{1412998609}}{982} x-\frac{37587}{982}=-\frac{\sqrt{1412998609}}{982}$$

Add \frac{37587}{982} to both sides of the equation.

$$x=\frac{\sqrt{1412998609}+37587}{982} x=\frac{37587-\sqrt{1412998609}}{982}$$