Solve 0.00391x^2+4.135x-2167.32=0

$$0.00391 { x }^{ 2 } +4.135x-2167.32=0$$

Solve for x

$$x = \frac{10 \sqrt{1274877745} - 206750}{391} \approx 384.409979265$$

$$x=\frac{-10\sqrt{1274877745}-206750}{391}\approx -1441.954736298$$

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.

$$0.00391x^{2}+4.135x-2167.32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.00391 for a, 4.135 for b, and -2167.32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4.135±\sqrt{4.135^{2}-4\times 0.00391\left(-2167.32\right)}}{2\times 0.00391}$$

Square 4.135 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-4.135±\sqrt{17.098225-4\times 0.00391\left(-2167.32\right)}}{2\times 0.00391}$$

Multiply -4 times 0.00391.

$$x=\frac{-4.135±\sqrt{17.098225-0.01564\left(-2167.32\right)}}{2\times 0.00391}$$

Multiply -0.01564 times -2167.32 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-4.135±\sqrt{17.098225+33.8968848}}{2\times 0.00391}$$

Add 17.098225 to 33.8968848 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-4.135±\sqrt{50.9951098}}{2\times 0.00391}$$

Take the square root of 50.9951098.

$$x=\frac{-4.135±\frac{\sqrt{1274877745}}{5000}}{2\times 0.00391}$$

Multiply 2 times 0.00391.

$$x=\frac{-4.135±\frac{\sqrt{1274877745}}{5000}}{0.00782}$$

Now solve the equation x=\frac{-4.135±\frac{\sqrt{1274877745}}{5000}}{0.00782} when ± is plus. Add -4.135 to \frac{\sqrt{1274877745}}{5000}.

$$x=\frac{\frac{\sqrt{1274877745}}{5000}-\frac{827}{200}}{0.00782}$$

Divide -\frac{827}{200}+\frac{\sqrt{1274877745}}{5000} by 0.00782 by multiplying -\frac{827}{200}+\frac{\sqrt{1274877745}}{5000} by the reciprocal of 0.00782.

$$x=\frac{10\sqrt{1274877745}-206750}{391}$$

Now solve the equation x=\frac{-4.135±\frac{\sqrt{1274877745}}{5000}}{0.00782} when ± is minus. Subtract \frac{\sqrt{1274877745}}{5000} from -4.135.

$$x=\frac{-\frac{\sqrt{1274877745}}{5000}-\frac{827}{200}}{0.00782}$$

Divide -\frac{827}{200}-\frac{\sqrt{1274877745}}{5000} by 0.00782 by multiplying -\frac{827}{200}-\frac{\sqrt{1274877745}}{5000} by the reciprocal of 0.00782.

$$x=\frac{-10\sqrt{1274877745}-206750}{391}$$

The equation is now solved.

$$x=\frac{10\sqrt{1274877745}-206750}{391} x=\frac{-10\sqrt{1274877745}-206750}{391}$$

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.

$$0.00391x^{2}+4.135x-2167.32=0$$

Add 2167.32 to both sides of the equation.

$$0.00391x^{2}+4.135x-2167.32-\left(-2167.32\right)=-\left(-2167.32\right)$$

Subtracting -2167.32 from itself leaves 0.

$$0.00391x^{2}+4.135x=-\left(-2167.32\right)$$

Subtract -2167.32 from 0.

$$0.00391x^{2}+4.135x=2167.32$$

Divide both sides of the equation by 0.00391, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.00391x^{2}+4.135x}{0.00391}=\frac{2167.32}{0.00391}$$

Dividing by 0.00391 undoes the multiplication by 0.00391.

$$x^{2}+\frac{4.135}{0.00391}x=\frac{2167.32}{0.00391}$$

Divide 4.135 by 0.00391 by multiplying 4.135 by the reciprocal of 0.00391.

$$x^{2}+\frac{413500}{391}x=\frac{2167.32}{0.00391}$$

Divide 2167.32 by 0.00391 by multiplying 2167.32 by the reciprocal of 0.00391.

$$x^{2}+\frac{413500}{391}x=\frac{216732000}{391}$$

Divide \frac{413500}{391}, the coefficient of the x term, by 2 to get \frac{206750}{391}. Then add the square of \frac{206750}{391} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{413500}{391}x+\frac{206750}{391}^{2}=\frac{216732000}{391}+\frac{206750}{391}^{2}$$

Square \frac{206750}{391} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{413500}{391}x+\frac{42745562500}{152881}=\frac{216732000}{391}+\frac{42745562500}{152881}$$

Add \frac{216732000}{391} to \frac{42745562500}{152881} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{413500}{391}x+\frac{42745562500}{152881}=\frac{127487774500}{152881}$$

Factor x^{2}+\frac{413500}{391}x+\frac{42745562500}{152881}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{206750}{391}\right)^{2}=\frac{127487774500}{152881}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{206750}{391}\right)^{2}}=\sqrt{\frac{127487774500}{152881}}$$

Simplify.

$$x+\frac{206750}{391}=\frac{10\sqrt{1274877745}}{391} x+\frac{206750}{391}=-\frac{10\sqrt{1274877745}}{391}$$

Subtract \frac{206750}{391} from both sides of the equation.

$$x=\frac{10\sqrt{1274877745}-206750}{391} x=\frac{-10\sqrt{1274877745}-206750}{391}$$