Solve x-9.8x^2/40000=0

$$x - \frac { 9.8 x ^ { 2 } } { 40000 } = 0$$

Solve for x

$$x = \frac{200000}{49} = 4081\frac{31}{49} \approx 4081.632653061$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 40000.

$$40000x-9.8x^{2}=0$$

Factor out x.

$$x\left(40000-9.8x\right)=0$$

To find equation solutions, solve x=0 and 40000-\frac{49x}{5}=0.

$$x=0 x=\frac{200000}{49}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 40000.

$$40000x-9.8x^{2}=0$$

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.

$$-9.8x^{2}+40000x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9.8 for a, 40000 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-40000±\sqrt{40000^{2}}}{2\left(-9.8\right)}$$

Take the square root of 40000^{2}.

$$x=\frac{-40000±40000}{2\left(-9.8\right)}$$

Multiply 2 times -9.8.

$$x=\frac{-40000±40000}{-19.6}$$

Now solve the equation x=\frac{-40000±40000}{-19.6} when ± is plus. Add -40000 to 40000.

$$x=\frac{0}{-19.6}$$

Divide 0 by -19.6 by multiplying 0 by the reciprocal of -19.6.

$$x=0$$

Now solve the equation x=\frac{-40000±40000}{-19.6} when ± is minus. Subtract 40000 from -40000.

$$x=-\frac{80000}{-19.6}$$

Divide -80000 by -19.6 by multiplying -80000 by the reciprocal of -19.6.

$$x=\frac{200000}{49}$$

The equation is now solved.

$$x=0 x=\frac{200000}{49}$$

Steps for Completing the Square

Multiply both sides of the equation by 40000.

$$40000x-9.8x^{2}=0$$

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.

$$-9.8x^{2}+40000x=0$$

Divide both sides of the equation by -9.8, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-9.8x^{2}+40000x}{-9.8}=\frac{0}{-9.8}$$

Dividing by -9.8 undoes the multiplication by -9.8.

$$x^{2}+\frac{40000}{-9.8}x=\frac{0}{-9.8}$$

Divide 40000 by -9.8 by multiplying 40000 by the reciprocal of -9.8.

$$x^{2}-\frac{200000}{49}x=\frac{0}{-9.8}$$

Divide 0 by -9.8 by multiplying 0 by the reciprocal of -9.8.

$$x^{2}-\frac{200000}{49}x=0$$

Divide -\frac{200000}{49}, the coefficient of the x term, by 2 to get -\frac{100000}{49}. Then add the square of -\frac{100000}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{200000}{49}x+\left(-\frac{100000}{49}\right)^{2}=\left(-\frac{100000}{49}\right)^{2}$$

Square -\frac{100000}{49} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{200000}{49}x+\frac{10000000000}{2401}=\frac{10000000000}{2401}$$

Factor x^{2}-\frac{200000}{49}x+\frac{10000000000}{2401}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{100000}{49}\right)^{2}=\frac{10000000000}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{100000}{49}\right)^{2}}=\sqrt{\frac{10000000000}{2401}}$$

Simplify.

$$x-\frac{100000}{49}=\frac{100000}{49} x-\frac{100000}{49}=-\frac{100000}{49}$$

Add \frac{100000}{49} to both sides of the equation.

$$x=\frac{200000}{49} x=0$$