Solve (90-30t)20t=50

$$( 90 - 30 t ) 20 t = 50$$

Solve for t

$$t=\frac{\sqrt{78}}{6}+\frac{3}{2}\approx 2.971960144$$

$$t=-\frac{\sqrt{78}}{6}+\frac{3}{2}\approx 0.028039856$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 90-30t by 20.

$$\left(1800-600t\right)t=50$$

Use the distributive property to multiply 1800-600t by t.

$$1800t-600t^{2}=50$$

Subtract 50 from both sides.

$$1800t-600t^{2}-50=0$$

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.

$$-600t^{2}+1800t-50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -600 for a, 1800 for b, and -50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-1800±\sqrt{1800^{2}-4\left(-600\right)\left(-50\right)}}{2\left(-600\right)}$$

Square 1800.

$$t=\frac{-1800±\sqrt{3240000-4\left(-600\right)\left(-50\right)}}{2\left(-600\right)}$$

Multiply -4 times -600.

$$t=\frac{-1800±\sqrt{3240000+2400\left(-50\right)}}{2\left(-600\right)}$$

Multiply 2400 times -50.

$$t=\frac{-1800±\sqrt{3240000-120000}}{2\left(-600\right)}$$

Add 3240000 to -120000.

$$t=\frac{-1800±\sqrt{3120000}}{2\left(-600\right)}$$

Take the square root of 3120000.

$$t=\frac{-1800±200\sqrt{78}}{2\left(-600\right)}$$

Multiply 2 times -600.

$$t=\frac{-1800±200\sqrt{78}}{-1200}$$

Now solve the equation t=\frac{-1800±200\sqrt{78}}{-1200} when ± is plus. Add -1800 to 200\sqrt{78}.

$$t=\frac{200\sqrt{78}-1800}{-1200}$$

Divide -1800+200\sqrt{78} by -1200.

$$t=-\frac{\sqrt{78}}{6}+\frac{3}{2}$$

Now solve the equation t=\frac{-1800±200\sqrt{78}}{-1200} when ± is minus. Subtract 200\sqrt{78} from -1800.

$$t=\frac{-200\sqrt{78}-1800}{-1200}$$

Divide -1800-200\sqrt{78} by -1200.

$$t=\frac{\sqrt{78}}{6}+\frac{3}{2}$$

The equation is now solved.

$$t=-\frac{\sqrt{78}}{6}+\frac{3}{2} t=\frac{\sqrt{78}}{6}+\frac{3}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 90-30t by 20.

$$\left(1800-600t\right)t=50$$

Use the distributive property to multiply 1800-600t by t.

$$1800t-600t^{2}=50$$

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.

$$-600t^{2}+1800t=50$$

Divide both sides by -600.

$$\frac{-600t^{2}+1800t}{-600}=\frac{50}{-600}$$

Dividing by -600 undoes the multiplication by -600.

$$t^{2}+\frac{1800}{-600}t=\frac{50}{-600}$$

Divide 1800 by -600.

$$t^{2}-3t=\frac{50}{-600}$$

Reduce the fraction \frac{50}{-600} to lowest terms by extracting and canceling out 50.

$$t^{2}-3t=-\frac{1}{12}$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-3t+\left(-\frac{3}{2}\right)^{2}=-\frac{1}{12}+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-3t+\frac{9}{4}=-\frac{1}{12}+\frac{9}{4}$$

Add -\frac{1}{12} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-3t+\frac{9}{4}=\frac{13}{6}$$

Factor t^{2}-3t+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{3}{2}\right)^{2}=\frac{13}{6}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{3}{2}\right)^{2}}=\sqrt{\frac{13}{6}}$$

Simplify.

$$t-\frac{3}{2}=\frac{\sqrt{78}}{6} t-\frac{3}{2}=-\frac{\sqrt{78}}{6}$$

Add \frac{3}{2} to both sides of the equation.

$$t=\frac{\sqrt{78}}{6}+\frac{3}{2} t=-\frac{\sqrt{78}}{6}+\frac{3}{2}$$