Solve 20x^2-15.84-39.2x=0

$$20 x ^ { 2 } - 15.84 - 39.2 x = 0$$

Solve for x

$$x = \frac{\sqrt{4381} + 49}{50} \approx 2.303782459$$

$$x=\frac{49-\sqrt{4381}}{50}\approx -0.343782459$$

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.

$$20x^{2}-39.2x-15.84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 20 for a, -39.2 for b, and -15.84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-39.2\right)±\sqrt{\left(-39.2\right)^{2}-4\times 20\left(-15.84\right)}}{2\times 20}$$

Square -39.2 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-39.2\right)±\sqrt{1536.64-4\times 20\left(-15.84\right)}}{2\times 20}$$

Multiply -4 times 20.

$$x=\frac{-\left(-39.2\right)±\sqrt{1536.64-80\left(-15.84\right)}}{2\times 20}$$

Multiply -80 times -15.84.

$$x=\frac{-\left(-39.2\right)±\sqrt{1536.64+1267.2}}{2\times 20}$$

Add 1536.64 to 1267.2 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-39.2\right)±\sqrt{2803.84}}{2\times 20}$$

Take the square root of 2803.84.

$$x=\frac{-\left(-39.2\right)±\frac{4\sqrt{4381}}{5}}{2\times 20}$$

The opposite of -39.2 is 39.2.

$$x=\frac{39.2±\frac{4\sqrt{4381}}{5}}{2\times 20}$$

Multiply 2 times 20.

$$x=\frac{39.2±\frac{4\sqrt{4381}}{5}}{40}$$

Now solve the equation x=\frac{39.2±\frac{4\sqrt{4381}}{5}}{40} when ± is plus. Add 39.2 to \frac{4\sqrt{4381}}{5}.

$$x=\frac{4\sqrt{4381}+196}{5\times 40}$$

Divide \frac{196+4\sqrt{4381}}{5} by 40.

$$x=\frac{\sqrt{4381}+49}{50}$$

Now solve the equation x=\frac{39.2±\frac{4\sqrt{4381}}{5}}{40} when ± is minus. Subtract \frac{4\sqrt{4381}}{5} from 39.2.

$$x=\frac{196-4\sqrt{4381}}{5\times 40}$$

Divide \frac{196-4\sqrt{4381}}{5} by 40.

$$x=\frac{49-\sqrt{4381}}{50}$$

The equation is now solved.

$$x=\frac{\sqrt{4381}+49}{50} x=\frac{49-\sqrt{4381}}{50}$$

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.

$$20x^{2}-39.2x-15.84=0$$

Add 15.84 to both sides of the equation.

$$20x^{2}-39.2x-15.84-\left(-15.84\right)=-\left(-15.84\right)$$

Subtracting -15.84 from itself leaves 0.

$$20x^{2}-39.2x=-\left(-15.84\right)$$

Subtract -15.84 from 0.

$$20x^{2}-39.2x=15.84$$

Divide both sides by 20.

$$\frac{20x^{2}-39.2x}{20}=\frac{15.84}{20}$$

Dividing by 20 undoes the multiplication by 20.

$$x^{2}+\left(-\frac{39.2}{20}\right)x=\frac{15.84}{20}$$

Divide -39.2 by 20.

$$x^{2}-1.96x=\frac{15.84}{20}$$

Divide 15.84 by 20.

$$x^{2}-1.96x=0.792$$

Divide -1.96, the coefficient of the x term, by 2 to get -0.98. Then add the square of -0.98 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1.96x+\left(-0.98\right)^{2}=0.792+\left(-0.98\right)^{2}$$

Square -0.98 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-1.96x+0.9604=0.792+0.9604$$

Add 0.792 to 0.9604 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-1.96x+0.9604=1.7524$$

Factor x^{2}-1.96x+0.9604. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.98\right)^{2}=1.7524$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.98\right)^{2}}=\sqrt{1.7524}$$

Simplify.

$$x-0.98=\frac{\sqrt{4381}}{50} x-0.98=-\frac{\sqrt{4381}}{50}$$

Add 0.98 to both sides of the equation.

$$x=\frac{\sqrt{4381}+49}{50} x=\frac{49-\sqrt{4381}}{50}$$