Solve 0.588+1.96x-24x^2=0

$$0.588 + 1.96 x - 24 x ^ { 2 } = 0$$

Solve for x

$$x=\frac{7\sqrt{769}+49}{1200}\approx 0.202596621$$

$$x=\frac{49-7\sqrt{769}}{1200}\approx -0.120929954$$

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.

$$-24x^{2}+1.96x+0.588=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -24 for a, 1.96 for b, and 0.588 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1.96±\sqrt{1.96^{2}-4\left(-24\right)\times 0.588}}{2\left(-24\right)}$$

Square 1.96 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-1.96±\sqrt{3.8416-4\left(-24\right)\times 0.588}}{2\left(-24\right)}$$

Multiply -4 times -24.

$$x=\frac{-1.96±\sqrt{3.8416+96\times 0.588}}{2\left(-24\right)}$$

Multiply 96 times 0.588.

$$x=\frac{-1.96±\sqrt{3.8416+56.448}}{2\left(-24\right)}$$

Add 3.8416 to 56.448 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1.96±\sqrt{60.2896}}{2\left(-24\right)}$$

Take the square root of 60.2896.

$$x=\frac{-1.96±\frac{7\sqrt{769}}{25}}{2\left(-24\right)}$$

Multiply 2 times -24.

$$x=\frac{-1.96±\frac{7\sqrt{769}}{25}}{-48}$$

Now solve the equation x=\frac{-1.96±\frac{7\sqrt{769}}{25}}{-48} when ± is plus. Add -1.96 to \frac{7\sqrt{769}}{25}.

$$x=\frac{7\sqrt{769}-49}{-48\times 25}$$

Divide \frac{-49+7\sqrt{769}}{25} by -48.

$$x=\frac{49-7\sqrt{769}}{1200}$$

Now solve the equation x=\frac{-1.96±\frac{7\sqrt{769}}{25}}{-48} when ± is minus. Subtract \frac{7\sqrt{769}}{25} from -1.96.

$$x=\frac{-7\sqrt{769}-49}{-48\times 25}$$

Divide \frac{-49-7\sqrt{769}}{25} by -48.

$$x=\frac{7\sqrt{769}+49}{1200}$$

The equation is now solved.

$$x=\frac{49-7\sqrt{769}}{1200} x=\frac{7\sqrt{769}+49}{1200}$$

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.

$$-24x^{2}+1.96x+0.588=0$$

Subtract 0.588 from both sides of the equation.

$$-24x^{2}+1.96x+0.588-0.588=-0.588$$

Subtracting 0.588 from itself leaves 0.

$$-24x^{2}+1.96x=-0.588$$

Divide both sides by -24.

$$\frac{-24x^{2}+1.96x}{-24}=-\frac{0.588}{-24}$$

Dividing by -24 undoes the multiplication by -24.

$$x^{2}+\frac{1.96}{-24}x=-\frac{0.588}{-24}$$

Divide 1.96 by -24.

$$x^{2}-\frac{49}{600}x=-\frac{0.588}{-24}$$

Divide -0.588 by -24.

$$x^{2}-\frac{49}{600}x=0.0245$$

Divide -\frac{49}{600}, the coefficient of the x term, by 2 to get -\frac{49}{1200}. Then add the square of -\frac{49}{1200} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{49}{600}x+\left(-\frac{49}{1200}\right)^{2}=0.0245+\left(-\frac{49}{1200}\right)^{2}$$

Square -\frac{49}{1200} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{49}{600}x+\frac{2401}{1440000}=0.0245+\frac{2401}{1440000}$$

Add 0.0245 to \frac{2401}{1440000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{49}{600}x+\frac{2401}{1440000}=\frac{37681}{1440000}$$

Factor x^{2}-\frac{49}{600}x+\frac{2401}{1440000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{49}{1200}\right)^{2}=\frac{37681}{1440000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{49}{1200}\right)^{2}}=\sqrt{\frac{37681}{1440000}}$$

Simplify.

$$x-\frac{49}{1200}=\frac{7\sqrt{769}}{1200} x-\frac{49}{1200}=-\frac{7\sqrt{769}}{1200}$$

Add \frac{49}{1200} to both sides of the equation.

$$x=\frac{7\sqrt{769}+49}{1200} x=\frac{49-7\sqrt{769}}{1200}$$