Solve 1.9=-0.0571x^2+0.383x+1.14

$$1.9 = -0.0571 { x }^{ 2 } +0.383x+1.14$$

Solve for x (complex solution)

$$x=\frac{1915+i\times 5\sqrt{26895}}{571}\approx 3.353765324+1.436050361i$$

$$x=\frac{-i\times 5\sqrt{26895}+1915}{571}\approx 3.353765324-1.436050361i$$

Steps Using the Quadratic Formula

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

$$-0.0571x^{2}+0.383x+1.14=1.9$$

Subtract 1.9 from both sides.

$$-0.0571x^{2}+0.383x+1.14-1.9=0$$

Subtract 1.9 from 1.14 to get -0.76.

$$-0.0571x^{2}+0.383x-0.76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.0571 for a, 0.383 for b, and -0.76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.383±\sqrt{0.383^{2}-4\left(-0.0571\right)\left(-0.76\right)}}{2\left(-0.0571\right)}$$

Square 0.383 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.383±\sqrt{0.146689-4\left(-0.0571\right)\left(-0.76\right)}}{2\left(-0.0571\right)}$$

Multiply -4 times -0.0571.

$$x=\frac{-0.383±\sqrt{0.146689+0.2284\left(-0.76\right)}}{2\left(-0.0571\right)}$$

Multiply 0.2284 times -0.76 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.383±\sqrt{0.146689-0.173584}}{2\left(-0.0571\right)}$$

Add 0.146689 to -0.173584 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.383±\sqrt{-0.026895}}{2\left(-0.0571\right)}$$

Take the square root of -0.026895.

$$x=\frac{-0.383±\frac{\sqrt{26895}i}{1000}}{2\left(-0.0571\right)}$$

Multiply 2 times -0.0571.

$$x=\frac{-0.383±\frac{\sqrt{26895}i}{1000}}{-0.1142}$$

Now solve the equation x=\frac{-0.383±\frac{\sqrt{26895}i}{1000}}{-0.1142} when ± is plus. Add -0.383 to \frac{i\sqrt{26895}}{1000}.

$$x=\frac{-383+\sqrt{26895}i}{-0.1142\times 1000}$$

Divide \frac{-383+i\sqrt{26895}}{1000} by -0.1142 by multiplying \frac{-383+i\sqrt{26895}}{1000} by the reciprocal of -0.1142.

$$x=\frac{-5\sqrt{26895}i+1915}{571}$$

Now solve the equation x=\frac{-0.383±\frac{\sqrt{26895}i}{1000}}{-0.1142} when ± is minus. Subtract \frac{i\sqrt{26895}}{1000} from -0.383.

$$x=\frac{-\sqrt{26895}i-383}{-0.1142\times 1000}$$

Divide \frac{-383-i\sqrt{26895}}{1000} by -0.1142 by multiplying \frac{-383-i\sqrt{26895}}{1000} by the reciprocal of -0.1142.

$$x=\frac{1915+5\sqrt{26895}i}{571}$$

The equation is now solved.

$$x=\frac{-5\sqrt{26895}i+1915}{571} x=\frac{1915+5\sqrt{26895}i}{571}$$

Steps for Completing the Square

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

$$-0.0571x^{2}+0.383x+1.14=1.9$$

Subtract 1.14 from both sides.

$$-0.0571x^{2}+0.383x=1.9-1.14$$

Subtract 1.14 from 1.9 to get 0.76.

$$-0.0571x^{2}+0.383x=0.76$$

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.

$$-0.0571x^{2}+0.383x=\frac{19}{25}$$

Divide both sides of the equation by -0.0571, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.0571x^{2}+0.383x}{-0.0571}=\frac{\frac{19}{25}}{-0.0571}$$

Dividing by -0.0571 undoes the multiplication by -0.0571.

$$x^{2}+\frac{0.383}{-0.0571}x=\frac{\frac{19}{25}}{-0.0571}$$

Divide 0.383 by -0.0571 by multiplying 0.383 by the reciprocal of -0.0571.

$$x^{2}-\frac{3830}{571}x=\frac{\frac{19}{25}}{-0.0571}$$

Divide \frac{19}{25} by -0.0571 by multiplying \frac{19}{25} by the reciprocal of -0.0571.

$$x^{2}-\frac{3830}{571}x=-\frac{7600}{571}$$

Divide -\frac{3830}{571}, the coefficient of the x term, by 2 to get -\frac{1915}{571}. Then add the square of -\frac{1915}{571} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3830}{571}x+\left(-\frac{1915}{571}\right)^{2}=-\frac{7600}{571}+\left(-\frac{1915}{571}\right)^{2}$$

Square -\frac{1915}{571} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3830}{571}x+\frac{3667225}{326041}=-\frac{7600}{571}+\frac{3667225}{326041}$$

Add -\frac{7600}{571} to \frac{3667225}{326041} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3830}{571}x+\frac{3667225}{326041}=-\frac{672375}{326041}$$

Factor x^{2}-\frac{3830}{571}x+\frac{3667225}{326041}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1915}{571}\right)^{2}=-\frac{672375}{326041}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1915}{571}\right)^{2}}=\sqrt{-\frac{672375}{326041}}$$

Simplify.

$$x-\frac{1915}{571}=\frac{5\sqrt{26895}i}{571} x-\frac{1915}{571}=-\frac{5\sqrt{26895}i}{571}$$

Add \frac{1915}{571} to both sides of the equation.

$$x=\frac{1915+5\sqrt{26895}i}{571} x=\frac{-5\sqrt{26895}i+1915}{571}$$