Solve 29w^2-30w=0

$$29 w ^ { 2 } - 30 w = 0$$

Solve for w

$$w = \frac{30}{29} = 1\frac{1}{29} \approx 1.034482759$$

$$w=0$$

Steps Using Factoring

Factor out w.

$$w\left(29w-30\right)=0$$

To find equation solutions, solve w=0 and 29w-30=0.

$$w=0 w=\frac{30}{29}$$

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.

$$29w^{2}-30w=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 29 for a, -30 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}}}{2\times 29}$$

Take the square root of \left(-30\right)^{2}.

$$w=\frac{-\left(-30\right)±30}{2\times 29}$$

The opposite of -30 is 30.

$$w=\frac{30±30}{2\times 29}$$

Multiply 2 times 29.

$$w=\frac{30±30}{58}$$

Now solve the equation w=\frac{30±30}{58} when ± is plus. Add 30 to 30.

$$w=\frac{60}{58}$$

Reduce the fraction \frac{60}{58} to lowest terms by extracting and canceling out 2.

$$w=\frac{30}{29}$$

Now solve the equation w=\frac{30±30}{58} when ± is minus. Subtract 30 from 30.

$$w=\frac{0}{58}$$

Divide 0 by 58.

$$w=0$$

The equation is now solved.

$$w=\frac{30}{29} w=0$$

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.

$$29w^{2}-30w=0$$

Divide both sides by 29.

$$\frac{29w^{2}-30w}{29}=\frac{0}{29}$$

Dividing by 29 undoes the multiplication by 29.

$$w^{2}-\frac{30}{29}w=\frac{0}{29}$$

Divide 0 by 29.

$$w^{2}-\frac{30}{29}w=0$$

Divide -\frac{30}{29}, the coefficient of the x term, by 2 to get -\frac{15}{29}. Then add the square of -\frac{15}{29} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}-\frac{30}{29}w+\left(-\frac{15}{29}\right)^{2}=\left(-\frac{15}{29}\right)^{2}$$

Square -\frac{15}{29} by squaring both the numerator and the denominator of the fraction.

$$w^{2}-\frac{30}{29}w+\frac{225}{841}=\frac{225}{841}$$

Factor w^{2}-\frac{30}{29}w+\frac{225}{841}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w-\frac{15}{29}\right)^{2}=\frac{225}{841}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-\frac{15}{29}\right)^{2}}=\sqrt{\frac{225}{841}}$$

Simplify.

$$w-\frac{15}{29}=\frac{15}{29} w-\frac{15}{29}=-\frac{15}{29}$$

Add \frac{15}{29} to both sides of the equation.

$$w=\frac{30}{29} w=0$$