Solve (t)=4+30t-16t^2

$$( t ) = 4 + 30 t - 16 t ^ { 2 }$$

Solve for t

$$t = \frac{\sqrt{1097} + 29}{32} \approx 1.941280948$$

$$t=\frac{29-\sqrt{1097}}{32}\approx -0.128780948$$

Steps Using the Quadratic Formula

Subtract 4 from both sides.

$$t-4=30t-16t^{2}$$

Subtract 30t from both sides.

$$t-4-30t=-16t^{2}$$

Combine t and -30t to get -29t.

$$-29t-4=-16t^{2}$$

Add 16t^{2} to both sides.

$$-29t-4+16t^{2}=0$$

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.

$$16t^{2}-29t-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -29 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-29\right)±\sqrt{\left(-29\right)^{2}-4\times 16\left(-4\right)}}{2\times 16}$$

Square -29.

$$t=\frac{-\left(-29\right)±\sqrt{841-4\times 16\left(-4\right)}}{2\times 16}$$

Multiply -4 times 16.

$$t=\frac{-\left(-29\right)±\sqrt{841-64\left(-4\right)}}{2\times 16}$$

Multiply -64 times -4.

$$t=\frac{-\left(-29\right)±\sqrt{841+256}}{2\times 16}$$

Add 841 to 256.

$$t=\frac{-\left(-29\right)±\sqrt{1097}}{2\times 16}$$

The opposite of -29 is 29.

$$t=\frac{29±\sqrt{1097}}{2\times 16}$$

Multiply 2 times 16.

$$t=\frac{29±\sqrt{1097}}{32}$$

Now solve the equation t=\frac{29±\sqrt{1097}}{32} when ± is plus. Add 29 to \sqrt{1097}.

$$t=\frac{\sqrt{1097}+29}{32}$$

Now solve the equation t=\frac{29±\sqrt{1097}}{32} when ± is minus. Subtract \sqrt{1097} from 29.

$$t=\frac{29-\sqrt{1097}}{32}$$

The equation is now solved.

$$t=\frac{\sqrt{1097}+29}{32} t=\frac{29-\sqrt{1097}}{32}$$

Steps for Completing the Square

Subtract 30t from both sides.

$$t-30t=4-16t^{2}$$

Combine t and -30t to get -29t.

$$-29t=4-16t^{2}$$

Add 16t^{2} to both sides.

$$-29t+16t^{2}=4$$

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.

$$16t^{2}-29t=4$$

Divide both sides by 16.

$$\frac{16t^{2}-29t}{16}=\frac{4}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$t^{2}-\frac{29}{16}t=\frac{4}{16}$$

Reduce the fraction \frac{4}{16} to lowest terms by extracting and canceling out 4.

$$t^{2}-\frac{29}{16}t=\frac{1}{4}$$

Divide -\frac{29}{16}, the coefficient of the x term, by 2 to get -\frac{29}{32}. Then add the square of -\frac{29}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{29}{16}t+\left(-\frac{29}{32}\right)^{2}=\frac{1}{4}+\left(-\frac{29}{32}\right)^{2}$$

Square -\frac{29}{32} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{29}{16}t+\frac{841}{1024}=\frac{1}{4}+\frac{841}{1024}$$

Add \frac{1}{4} to \frac{841}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{29}{16}t+\frac{841}{1024}=\frac{1097}{1024}$$

Factor t^{2}-\frac{29}{16}t+\frac{841}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{29}{32}\right)^{2}=\frac{1097}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{29}{32}\right)^{2}}=\sqrt{\frac{1097}{1024}}$$

Simplify.

$$t-\frac{29}{32}=\frac{\sqrt{1097}}{32} t-\frac{29}{32}=-\frac{\sqrt{1097}}{32}$$

Add \frac{29}{32} to both sides of the equation.

$$t=\frac{\sqrt{1097}+29}{32} t=\frac{29-\sqrt{1097}}{32}$$