Solve 33j^2-16j=0

$$33 j ^ { 2 } - 16 j = 0$$

Solve for j

$$j=\frac{16}{33}\approx 0.484848485$$

$$j=0$$

Steps Using Factoring

Factor out j.

$$j\left(33j-16\right)=0$$

To find equation solutions, solve j=0 and 33j-16=0.

$$j=0 j=\frac{16}{33}$$

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.

$$33j^{2}-16j=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 33 for a, -16 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$j=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}}}{2\times 33}$$

Take the square root of \left(-16\right)^{2}.

$$j=\frac{-\left(-16\right)±16}{2\times 33}$$

The opposite of -16 is 16.

$$j=\frac{16±16}{2\times 33}$$

Multiply 2 times 33.

$$j=\frac{16±16}{66}$$

Now solve the equation j=\frac{16±16}{66} when ± is plus. Add 16 to 16.

$$j=\frac{32}{66}$$

Reduce the fraction \frac{32}{66} to lowest terms by extracting and canceling out 2.

$$j=\frac{16}{33}$$

Now solve the equation j=\frac{16±16}{66} when ± is minus. Subtract 16 from 16.

$$j=\frac{0}{66}$$

Divide 0 by 66.

$$j=0$$

The equation is now solved.

$$j=\frac{16}{33} j=0$$

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.

$$33j^{2}-16j=0$$

Divide both sides by 33.

$$\frac{33j^{2}-16j}{33}=\frac{0}{33}$$

Dividing by 33 undoes the multiplication by 33.

$$j^{2}-\frac{16}{33}j=\frac{0}{33}$$

Divide 0 by 33.

$$j^{2}-\frac{16}{33}j=0$$

Divide -\frac{16}{33}, the coefficient of the x term, by 2 to get -\frac{8}{33}. Then add the square of -\frac{8}{33} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$j^{2}-\frac{16}{33}j+\left(-\frac{8}{33}\right)^{2}=\left(-\frac{8}{33}\right)^{2}$$

Square -\frac{8}{33} by squaring both the numerator and the denominator of the fraction.

$$j^{2}-\frac{16}{33}j+\frac{64}{1089}=\frac{64}{1089}$$

Factor j^{2}-\frac{16}{33}j+\frac{64}{1089}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(j-\frac{8}{33}\right)^{2}=\frac{64}{1089}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(j-\frac{8}{33}\right)^{2}}=\sqrt{\frac{64}{1089}}$$

Simplify.

$$j-\frac{8}{33}=\frac{8}{33} j-\frac{8}{33}=-\frac{8}{33}$$

Add \frac{8}{33} to both sides of the equation.

$$j=\frac{16}{33} j=0$$