Solve 10t(5t-1)=0

$$10 t ( 5 t - 1 ) = 0$$

Solve for t

$$t=\frac{1}{5}=0.2$$

$$t=0$$

Steps Using Factoring

Use the distributive property to multiply 10t by 5t-1.

$$50t^{2}-10t=0$$

Factor out t.

$$t\left(50t-10\right)=0$$

To find equation solutions, solve t=0 and 50t-10=0.

$$t=0 t=\frac{1}{5}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10t by 5t-1.

$$50t^{2}-10t=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 50 for a, -10 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}}}{2\times 50}$$

Take the square root of \left(-10\right)^{2}.

$$t=\frac{-\left(-10\right)±10}{2\times 50}$$

The opposite of -10 is 10.

$$t=\frac{10±10}{2\times 50}$$

Multiply 2 times 50.

$$t=\frac{10±10}{100}$$

Now solve the equation t=\frac{10±10}{100} when ± is plus. Add 10 to 10.

$$t=\frac{20}{100}$$

Reduce the fraction \frac{20}{100} to lowest terms by extracting and canceling out 20.

$$t=\frac{1}{5}$$

Now solve the equation t=\frac{10±10}{100} when ± is minus. Subtract 10 from 10.

$$t=\frac{0}{100}$$

Divide 0 by 100.

$$t=0$$

The equation is now solved.

$$t=\frac{1}{5} t=0$$

Steps for Completing the Square

Use the distributive property to multiply 10t by 5t-1.

$$50t^{2}-10t=0$$

Divide both sides by 50.

$$\frac{50t^{2}-10t}{50}=\frac{0}{50}$$

Dividing by 50 undoes the multiplication by 50.

$$t^{2}+\left(-\frac{10}{50}\right)t=\frac{0}{50}$$

Reduce the fraction \frac{-10}{50} to lowest terms by extracting and canceling out 10.

$$t^{2}-\frac{1}{5}t=\frac{0}{50}$$

Divide 0 by 50.

$$t^{2}-\frac{1}{5}t=0$$

Divide -\frac{1}{5}, the coefficient of the x term, by 2 to get -\frac{1}{10}. Then add the square of -\frac{1}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{1}{5}t+\left(-\frac{1}{10}\right)^{2}=\left(-\frac{1}{10}\right)^{2}$$

Square -\frac{1}{10} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{1}{5}t+\frac{1}{100}=\frac{1}{100}$$

Factor t^{2}-\frac{1}{5}t+\frac{1}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{1}{10}\right)^{2}=\frac{1}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{1}{10}\right)^{2}}=\sqrt{\frac{1}{100}}$$

Simplify.

$$t-\frac{1}{10}=\frac{1}{10} t-\frac{1}{10}=-\frac{1}{10}$$

Add \frac{1}{10} to both sides of the equation.

$$t=\frac{1}{5} t=0$$