Solve 8.314x+0.0149x^2-1000=0

$$8.314x+0.0149 { x }^{ 2 } -1000 = 0$$

Solve for x

$$x = \frac{10 \sqrt{32180649} - 41570}{149} \approx 101.731479501$$

$$x=\frac{-10\sqrt{32180649}-41570}{149}\approx -659.718056682$$

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.

$$0.0149x^{2}+8.314x-1000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.0149 for a, 8.314 for b, and -1000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8.314±\sqrt{8.314^{2}-4\times 0.0149\left(-1000\right)}}{2\times 0.0149}$$

Square 8.314 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-8.314±\sqrt{69.122596-4\times 0.0149\left(-1000\right)}}{2\times 0.0149}$$

Multiply -4 times 0.0149.

$$x=\frac{-8.314±\sqrt{69.122596-0.0596\left(-1000\right)}}{2\times 0.0149}$$

Multiply -0.0596 times -1000.

$$x=\frac{-8.314±\sqrt{69.122596+59.6}}{2\times 0.0149}$$

Add 69.122596 to 59.6 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-8.314±\sqrt{128.722596}}{2\times 0.0149}$$

Take the square root of 128.722596.

$$x=\frac{-8.314±\frac{\sqrt{32180649}}{500}}{2\times 0.0149}$$

Multiply 2 times 0.0149.

$$x=\frac{-8.314±\frac{\sqrt{32180649}}{500}}{0.0298}$$

Now solve the equation x=\frac{-8.314±\frac{\sqrt{32180649}}{500}}{0.0298} when ± is plus. Add -8.314 to \frac{\sqrt{32180649}}{500}.

$$x=\frac{\sqrt{32180649}-4157}{0.0298\times 500}$$

Divide \frac{-4157+\sqrt{32180649}}{500} by 0.0298 by multiplying \frac{-4157+\sqrt{32180649}}{500} by the reciprocal of 0.0298.

$$x=\frac{10\sqrt{32180649}-41570}{149}$$

Now solve the equation x=\frac{-8.314±\frac{\sqrt{32180649}}{500}}{0.0298} when ± is minus. Subtract \frac{\sqrt{32180649}}{500} from -8.314.

$$x=\frac{-\sqrt{32180649}-4157}{0.0298\times 500}$$

Divide \frac{-4157-\sqrt{32180649}}{500} by 0.0298 by multiplying \frac{-4157-\sqrt{32180649}}{500} by the reciprocal of 0.0298.

$$x=\frac{-10\sqrt{32180649}-41570}{149}$$

The equation is now solved.

$$x=\frac{10\sqrt{32180649}-41570}{149} x=\frac{-10\sqrt{32180649}-41570}{149}$$

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.

$$0.0149x^{2}+8.314x-1000=0$$

Add 1000 to both sides of the equation.

$$0.0149x^{2}+8.314x-1000-\left(-1000\right)=-\left(-1000\right)$$

Subtracting -1000 from itself leaves 0.

$$0.0149x^{2}+8.314x=-\left(-1000\right)$$

Subtract -1000 from 0.

$$0.0149x^{2}+8.314x=1000$$

Divide both sides of the equation by 0.0149, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.0149x^{2}+8.314x}{0.0149}=\frac{1000}{0.0149}$$

Dividing by 0.0149 undoes the multiplication by 0.0149.

$$x^{2}+\frac{8.314}{0.0149}x=\frac{1000}{0.0149}$$

Divide 8.314 by 0.0149 by multiplying 8.314 by the reciprocal of 0.0149.

$$x^{2}+\frac{83140}{149}x=\frac{1000}{0.0149}$$

Divide 1000 by 0.0149 by multiplying 1000 by the reciprocal of 0.0149.

$$x^{2}+\frac{83140}{149}x=\frac{10000000}{149}$$

Divide \frac{83140}{149}, the coefficient of the x term, by 2 to get \frac{41570}{149}. Then add the square of \frac{41570}{149} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{83140}{149}x+\frac{41570}{149}^{2}=\frac{10000000}{149}+\frac{41570}{149}^{2}$$

Square \frac{41570}{149} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{83140}{149}x+\frac{1728064900}{22201}=\frac{10000000}{149}+\frac{1728064900}{22201}$$

Add \frac{10000000}{149} to \frac{1728064900}{22201} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{83140}{149}x+\frac{1728064900}{22201}=\frac{3218064900}{22201}$$

Factor x^{2}+\frac{83140}{149}x+\frac{1728064900}{22201}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{41570}{149}\right)^{2}=\frac{3218064900}{22201}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{41570}{149}\right)^{2}}=\sqrt{\frac{3218064900}{22201}}$$

Simplify.

$$x+\frac{41570}{149}=\frac{10\sqrt{32180649}}{149} x+\frac{41570}{149}=-\frac{10\sqrt{32180649}}{149}$$

Subtract \frac{41570}{149} from both sides of the equation.

$$x=\frac{10\sqrt{32180649}-41570}{149} x=\frac{-10\sqrt{32180649}-41570}{149}$$