Solve 0=34x-9.81x^2/2

$$0=34x- \frac{ 9.81 { x }^{ 2 } }{ 2 }$$

Solve for x

$$x = \frac{6800}{981} = 6\frac{914}{981} \approx 6.931702345$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 2.

$$0=68x-9.81x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$68x-9.81x^{2}=0$$

Factor out x.

$$x\left(68-9.81x\right)=0$$

To find equation solutions, solve x=0 and 68-\frac{981x}{100}=0.

$$x=0 x=\frac{6800}{981}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$0=68x-9.81x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$68x-9.81x^{2}=0$$

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.

$$-9.81x^{2}+68x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9.81 for a, 68 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-68±\sqrt{68^{2}}}{2\left(-9.81\right)}$$

Take the square root of 68^{2}.

$$x=\frac{-68±68}{2\left(-9.81\right)}$$

Multiply 2 times -9.81.

$$x=\frac{-68±68}{-19.62}$$

Now solve the equation x=\frac{-68±68}{-19.62} when ± is plus. Add -68 to 68.

$$x=\frac{0}{-19.62}$$

Divide 0 by -19.62 by multiplying 0 by the reciprocal of -19.62.

$$x=0$$

Now solve the equation x=\frac{-68±68}{-19.62} when ± is minus. Subtract 68 from -68.

$$x=-\frac{136}{-19.62}$$

Divide -136 by -19.62 by multiplying -136 by the reciprocal of -19.62.

$$x=\frac{6800}{981}$$

The equation is now solved.

$$x=0 x=\frac{6800}{981}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$0=68x-9.81x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$68x-9.81x^{2}=0$$

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.

$$-9.81x^{2}+68x=0$$

Divide both sides of the equation by -9.81, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-9.81x^{2}+68x}{-9.81}=\frac{0}{-9.81}$$

Dividing by -9.81 undoes the multiplication by -9.81.

$$x^{2}+\frac{68}{-9.81}x=\frac{0}{-9.81}$$

Divide 68 by -9.81 by multiplying 68 by the reciprocal of -9.81.

$$x^{2}-\frac{6800}{981}x=\frac{0}{-9.81}$$

Divide 0 by -9.81 by multiplying 0 by the reciprocal of -9.81.

$$x^{2}-\frac{6800}{981}x=0$$

Divide -\frac{6800}{981}, the coefficient of the x term, by 2 to get -\frac{3400}{981}. Then add the square of -\frac{3400}{981} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{6800}{981}x+\left(-\frac{3400}{981}\right)^{2}=\left(-\frac{3400}{981}\right)^{2}$$

Square -\frac{3400}{981} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{6800}{981}x+\frac{11560000}{962361}=\frac{11560000}{962361}$$

Factor x^{2}-\frac{6800}{981}x+\frac{11560000}{962361}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3400}{981}\right)^{2}=\frac{11560000}{962361}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3400}{981}\right)^{2}}=\sqrt{\frac{11560000}{962361}}$$

Simplify.

$$x-\frac{3400}{981}=\frac{3400}{981} x-\frac{3400}{981}=-\frac{3400}{981}$$

Add \frac{3400}{981} to both sides of the equation.

$$x=\frac{6800}{981} x=0$$