Solve 0=100+30t-5t^2

$$0 = 100 + 30 t - 5 t ^ { 2 }$$

Solve for t

$$t=\sqrt{29}+3\approx 8.385164807$$

$$t=3-\sqrt{29}\approx -2.385164807$$

Steps Using the Quadratic Formula

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

$$100+30t-5t^{2}=0$$

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.

$$-5t^{2}+30t+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 30 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-30±\sqrt{30^{2}-4\left(-5\right)\times 100}}{2\left(-5\right)}$$

Square 30.

$$t=\frac{-30±\sqrt{900-4\left(-5\right)\times 100}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$t=\frac{-30±\sqrt{900+20\times 100}}{2\left(-5\right)}$$

Multiply 20 times 100.

$$t=\frac{-30±\sqrt{900+2000}}{2\left(-5\right)}$$

Add 900 to 2000.

$$t=\frac{-30±\sqrt{2900}}{2\left(-5\right)}$$

Take the square root of 2900.

$$t=\frac{-30±10\sqrt{29}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$t=\frac{-30±10\sqrt{29}}{-10}$$

Now solve the equation t=\frac{-30±10\sqrt{29}}{-10} when ± is plus. Add -30 to 10\sqrt{29}.

$$t=\frac{10\sqrt{29}-30}{-10}$$

Divide -30+10\sqrt{29} by -10.

$$t=3-\sqrt{29}$$

Now solve the equation t=\frac{-30±10\sqrt{29}}{-10} when ± is minus. Subtract 10\sqrt{29} from -30.

$$t=\frac{-10\sqrt{29}-30}{-10}$$

Divide -30-10\sqrt{29} by -10.

$$t=\sqrt{29}+3$$

The equation is now solved.

$$t=3-\sqrt{29} t=\sqrt{29}+3$$

Steps for Completing the Square

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

$$100+30t-5t^{2}=0$$

Subtract 100 from both sides. Anything subtracted from zero gives its negation.

$$30t-5t^{2}=-100$$

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.

$$-5t^{2}+30t=-100$$

Divide both sides by -5.

$$\frac{-5t^{2}+30t}{-5}=-\frac{100}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$t^{2}+\frac{30}{-5}t=-\frac{100}{-5}$$

Divide 30 by -5.

$$t^{2}-6t=-\frac{100}{-5}$$

Divide -100 by -5.

$$t^{2}-6t=20$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-6t+\left(-3\right)^{2}=20+\left(-3\right)^{2}$$

Square -3.

$$t^{2}-6t+9=20+9$$

Add 20 to 9.

$$t^{2}-6t+9=29$$

Factor t^{2}-6t+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-3\right)^{2}=29$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-3\right)^{2}}=\sqrt{29}$$

Simplify.

$$t-3=\sqrt{29} t-3=-\sqrt{29}$$

Add 3 to both sides of the equation.

$$t=\sqrt{29}+3 t=3-\sqrt{29}$$