Solve x=3(1--20x/-0.5)^2

$$x = 3 ( 1 - \frac { - 20 x } { - 0.5 } ) ^ { 2 }$$

Solve for x

$$x=\frac{\sqrt{481}+241}{9600}\approx 0.02738872$$

$$x=\frac{241-\sqrt{481}}{9600}\approx 0.022819613$$

Steps Using the Quadratic Formula

Divide -20x by -0.5 to get 40x.

$$x=3\left(1-40x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1-40x\right)^{2}.

$$x=3\left(1+2\left(-40x\right)+\left(-40x\right)^{2}\right)$$

Calculate -40x to the power of 2 and get \left(40x\right)^{2}.

$$x=3\left(1+2\left(-40x\right)+\left(40x\right)^{2}\right)$$

Expand \left(40x\right)^{2}.

$$x=3\left(1+2\left(-40x\right)+40^{2}x^{2}\right)$$

Calculate 40 to the power of 2 and get 1600.

$$x=3\left(1+2\left(-40x\right)+1600x^{2}\right)$$

Use the distributive property to multiply 3 by 1+2\left(-40x\right)+1600x^{2}.

$$x=3+6\left(-40x\right)+4800x^{2}$$

Subtract 3 from both sides.

$$x-3=6\left(-40x\right)+4800x^{2}$$

Subtract 6\left(-40x\right) from both sides.

$$x-3-6\left(-40x\right)=4800x^{2}$$

Subtract 4800x^{2} from both sides.

$$x-3-6\left(-40x\right)-4800x^{2}=0$$

Multiply -1 and 6 to get -6.

$$x-3-6\left(-1\right)\times 40x-4800x^{2}=0$$

Multiply -6 and -1 to get 6.

$$x-3+6\times 40x-4800x^{2}=0$$

Multiply 6 and 40 to get 240.

$$x-3+240x-4800x^{2}=0$$

Combine x and 240x to get 241x.

$$241x-3-4800x^{2}=0$$

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.

$$-4800x^{2}+241x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4800 for a, 241 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-241±\sqrt{241^{2}-4\left(-4800\right)\left(-3\right)}}{2\left(-4800\right)}$$

Square 241.

$$x=\frac{-241±\sqrt{58081-4\left(-4800\right)\left(-3\right)}}{2\left(-4800\right)}$$

Multiply -4 times -4800.

$$x=\frac{-241±\sqrt{58081+19200\left(-3\right)}}{2\left(-4800\right)}$$

Multiply 19200 times -3.

$$x=\frac{-241±\sqrt{58081-57600}}{2\left(-4800\right)}$$

Add 58081 to -57600.

$$x=\frac{-241±\sqrt{481}}{2\left(-4800\right)}$$

Multiply 2 times -4800.

$$x=\frac{-241±\sqrt{481}}{-9600}$$

Now solve the equation x=\frac{-241±\sqrt{481}}{-9600} when ± is plus. Add -241 to \sqrt{481}.

$$x=\frac{\sqrt{481}-241}{-9600}$$

Divide -241+\sqrt{481} by -9600.

$$x=\frac{241-\sqrt{481}}{9600}$$

Now solve the equation x=\frac{-241±\sqrt{481}}{-9600} when ± is minus. Subtract \sqrt{481} from -241.

$$x=\frac{-\sqrt{481}-241}{-9600}$$

Divide -241-\sqrt{481} by -9600.

$$x=\frac{\sqrt{481}+241}{9600}$$

The equation is now solved.

$$x=\frac{241-\sqrt{481}}{9600} x=\frac{\sqrt{481}+241}{9600}$$

Steps for Completing the Square

Divide -20x by -0.5 to get 40x.

$$x=3\left(1-40x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1-40x\right)^{2}.

$$x=3\left(1+2\left(-40x\right)+\left(-40x\right)^{2}\right)$$

Calculate -40x to the power of 2 and get \left(40x\right)^{2}.

$$x=3\left(1+2\left(-40x\right)+\left(40x\right)^{2}\right)$$

Expand \left(40x\right)^{2}.

$$x=3\left(1+2\left(-40x\right)+40^{2}x^{2}\right)$$

Calculate 40 to the power of 2 and get 1600.

$$x=3\left(1+2\left(-40x\right)+1600x^{2}\right)$$

Use the distributive property to multiply 3 by 1+2\left(-40x\right)+1600x^{2}.

$$x=3+6\left(-40x\right)+4800x^{2}$$

Subtract 6\left(-40x\right) from both sides.

$$x-6\left(-40x\right)=3+4800x^{2}$$

Subtract 4800x^{2} from both sides.

$$x-6\left(-40x\right)-4800x^{2}=3$$

Multiply -1 and 6 to get -6.

$$x-6\left(-1\right)\times 40x-4800x^{2}=3$$

Multiply -6 and -1 to get 6.

$$x+6\times 40x-4800x^{2}=3$$

Multiply 6 and 40 to get 240.

$$x+240x-4800x^{2}=3$$

Combine x and 240x to get 241x.

$$241x-4800x^{2}=3$$

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.

$$-4800x^{2}+241x=3$$

Divide both sides by -4800.

$$\frac{-4800x^{2}+241x}{-4800}=\frac{3}{-4800}$$

Dividing by -4800 undoes the multiplication by -4800.

$$x^{2}+\frac{241}{-4800}x=\frac{3}{-4800}$$

Divide 241 by -4800.

$$x^{2}-\frac{241}{4800}x=\frac{3}{-4800}$$

Reduce the fraction \frac{3}{-4800} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{241}{4800}x=-\frac{1}{1600}$$

Divide -\frac{241}{4800}, the coefficient of the x term, by 2 to get -\frac{241}{9600}. Then add the square of -\frac{241}{9600} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{241}{4800}x+\left(-\frac{241}{9600}\right)^{2}=-\frac{1}{1600}+\left(-\frac{241}{9600}\right)^{2}$$

Square -\frac{241}{9600} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{241}{4800}x+\frac{58081}{92160000}=-\frac{1}{1600}+\frac{58081}{92160000}$$

Add -\frac{1}{1600} to \frac{58081}{92160000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{241}{4800}x+\frac{58081}{92160000}=\frac{481}{92160000}$$

Factor x^{2}-\frac{241}{4800}x+\frac{58081}{92160000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{241}{9600}\right)^{2}=\frac{481}{92160000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{241}{9600}\right)^{2}}=\sqrt{\frac{481}{92160000}}$$

Simplify.

$$x-\frac{241}{9600}=\frac{\sqrt{481}}{9600} x-\frac{241}{9600}=-\frac{\sqrt{481}}{9600}$$

Add \frac{241}{9600} to both sides of the equation.

$$x=\frac{\sqrt{481}+241}{9600} x=\frac{241-\sqrt{481}}{9600}$$