Solve 3.9248x^2+0.724x-0.6087=0

$$3.9248 x ^ { 2 } + 0.724 x - 0.6087 = 0$$

Solve for x

$$x=\frac{\sqrt{3937609}}{4906}-\frac{905}{9812}\approx 0.312238269$$

$$x=-\frac{\sqrt{3937609}}{4906}-\frac{905}{9812}\approx -0.496706268$$

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.

$$3.9248x^{2}+0.724x-0.6087=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3.9248 for a, 0.724 for b, and -0.6087 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.724±\sqrt{0.724^{2}-4\times 3.9248\left(-0.6087\right)}}{2\times 3.9248}$$

Square 0.724 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.724±\sqrt{0.524176-4\times 3.9248\left(-0.6087\right)}}{2\times 3.9248}$$

Multiply -4 times 3.9248.

$$x=\frac{-0.724±\sqrt{0.524176-15.6992\left(-0.6087\right)}}{2\times 3.9248}$$

Multiply -15.6992 times -0.6087 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.724±\sqrt{0.524176+9.55610304}}{2\times 3.9248}$$

Add 0.524176 to 9.55610304 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.724±\sqrt{10.08027904}}{2\times 3.9248}$$

Take the square root of 10.08027904.

$$x=\frac{-0.724±\frac{\sqrt{3937609}}{625}}{2\times 3.9248}$$

Multiply 2 times 3.9248.

$$x=\frac{-0.724±\frac{\sqrt{3937609}}{625}}{7.8496}$$

Now solve the equation x=\frac{-0.724±\frac{\sqrt{3937609}}{625}}{7.8496} when ± is plus. Add -0.724 to \frac{\sqrt{3937609}}{625}.

$$x=\frac{\frac{\sqrt{3937609}}{625}-\frac{181}{250}}{7.8496}$$

Divide -\frac{181}{250}+\frac{\sqrt{3937609}}{625} by 7.8496 by multiplying -\frac{181}{250}+\frac{\sqrt{3937609}}{625} by the reciprocal of 7.8496.

$$x=\frac{\sqrt{3937609}}{4906}-\frac{905}{9812}$$

Now solve the equation x=\frac{-0.724±\frac{\sqrt{3937609}}{625}}{7.8496} when ± is minus. Subtract \frac{\sqrt{3937609}}{625} from -0.724.

$$x=\frac{-\frac{\sqrt{3937609}}{625}-\frac{181}{250}}{7.8496}$$

Divide -\frac{181}{250}-\frac{\sqrt{3937609}}{625} by 7.8496 by multiplying -\frac{181}{250}-\frac{\sqrt{3937609}}{625} by the reciprocal of 7.8496.

$$x=-\frac{\sqrt{3937609}}{4906}-\frac{905}{9812}$$

The equation is now solved.

$$x=\frac{\sqrt{3937609}}{4906}-\frac{905}{9812} x=-\frac{\sqrt{3937609}}{4906}-\frac{905}{9812}$$

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.

$$3.9248x^{2}+0.724x-0.6087=0$$

Add 0.6087 to both sides of the equation.

$$3.9248x^{2}+0.724x-0.6087-\left(-0.6087\right)=-\left(-0.6087\right)$$

Subtracting -0.6087 from itself leaves 0.

$$3.9248x^{2}+0.724x=-\left(-0.6087\right)$$

Subtract -0.6087 from 0.

$$3.9248x^{2}+0.724x=0.6087$$

Divide both sides of the equation by 3.9248, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{3.9248x^{2}+0.724x}{3.9248}=\frac{0.6087}{3.9248}$$

Dividing by 3.9248 undoes the multiplication by 3.9248.

$$x^{2}+\frac{0.724}{3.9248}x=\frac{0.6087}{3.9248}$$

Divide 0.724 by 3.9248 by multiplying 0.724 by the reciprocal of 3.9248.

$$x^{2}+\frac{905}{4906}x=\frac{0.6087}{3.9248}$$

Divide 0.6087 by 3.9248 by multiplying 0.6087 by the reciprocal of 3.9248.

$$x^{2}+\frac{905}{4906}x=\frac{6087}{39248}$$

Divide \frac{905}{4906}, the coefficient of the x term, by 2 to get \frac{905}{9812}. Then add the square of \frac{905}{9812} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{905}{4906}x+\frac{905}{9812}^{2}=\frac{6087}{39248}+\frac{905}{9812}^{2}$$

Square \frac{905}{9812} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{905}{4906}x+\frac{819025}{96275344}=\frac{6087}{39248}+\frac{819025}{96275344}$$

Add \frac{6087}{39248} to \frac{819025}{96275344} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{905}{4906}x+\frac{819025}{96275344}=\frac{3937609}{24068836}$$

Factor x^{2}+\frac{905}{4906}x+\frac{819025}{96275344}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{905}{9812}\right)^{2}=\frac{3937609}{24068836}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{905}{9812}\right)^{2}}=\sqrt{\frac{3937609}{24068836}}$$

Simplify.

$$x+\frac{905}{9812}=\frac{\sqrt{3937609}}{4906} x+\frac{905}{9812}=-\frac{\sqrt{3937609}}{4906}$$

Subtract \frac{905}{9812} from both sides of the equation.

$$x=\frac{\sqrt{3937609}}{4906}-\frac{905}{9812} x=-\frac{\sqrt{3937609}}{4906}-\frac{905}{9812}$$