Solve t=16t^2+48t+20

$$t = 16 t ^ { 2 } + 48 t + 20$$

Solve for t

$$t=\frac{\sqrt{929}-47}{32}\approx -0.516265584$$

$$t=\frac{-\sqrt{929}-47}{32}\approx -2.421234416$$

Steps Using the Quadratic Formula

Subtract 16t^{2} from both sides.

$$t-16t^{2}=48t+20$$

Subtract 48t from both sides.

$$t-16t^{2}-48t=20$$

Combine t and -48t to get -47t.

$$-47t-16t^{2}=20$$

Subtract 20 from both sides.

$$-47t-16t^{2}-20=0$$

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.

$$-16t^{2}-47t-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, -47 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-47\right)±\sqrt{\left(-47\right)^{2}-4\left(-16\right)\left(-20\right)}}{2\left(-16\right)}$$

Square -47.

$$t=\frac{-\left(-47\right)±\sqrt{2209-4\left(-16\right)\left(-20\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-\left(-47\right)±\sqrt{2209+64\left(-20\right)}}{2\left(-16\right)}$$

Multiply 64 times -20.

$$t=\frac{-\left(-47\right)±\sqrt{2209-1280}}{2\left(-16\right)}$$

Add 2209 to -1280.

$$t=\frac{-\left(-47\right)±\sqrt{929}}{2\left(-16\right)}$$

The opposite of -47 is 47.

$$t=\frac{47±\sqrt{929}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{47±\sqrt{929}}{-32}$$

Now solve the equation t=\frac{47±\sqrt{929}}{-32} when ± is plus. Add 47 to \sqrt{929}.

$$t=\frac{\sqrt{929}+47}{-32}$$

Divide 47+\sqrt{929} by -32.

$$t=\frac{-\sqrt{929}-47}{32}$$

Now solve the equation t=\frac{47±\sqrt{929}}{-32} when ± is minus. Subtract \sqrt{929} from 47.

$$t=\frac{47-\sqrt{929}}{-32}$$

Divide 47-\sqrt{929} by -32.

$$t=\frac{\sqrt{929}-47}{32}$$

The equation is now solved.

$$t=\frac{-\sqrt{929}-47}{32} t=\frac{\sqrt{929}-47}{32}$$

Steps for Completing the Square

Subtract 16t^{2} from both sides.

$$t-16t^{2}=48t+20$$

Subtract 48t from both sides.

$$t-16t^{2}-48t=20$$

Combine t and -48t to get -47t.

$$-47t-16t^{2}=20$$

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.

$$-16t^{2}-47t=20$$

Divide both sides by -16.

$$\frac{-16t^{2}-47t}{-16}=\frac{20}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$t^{2}+\left(-\frac{47}{-16}\right)t=\frac{20}{-16}$$

Divide -47 by -16.

$$t^{2}+\frac{47}{16}t=\frac{20}{-16}$$

Reduce the fraction \frac{20}{-16} to lowest terms by extracting and canceling out 4.

$$t^{2}+\frac{47}{16}t=-\frac{5}{4}$$

Divide \frac{47}{16}, the coefficient of the x term, by 2 to get \frac{47}{32}. Then add the square of \frac{47}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{47}{16}t+\left(\frac{47}{32}\right)^{2}=-\frac{5}{4}+\left(\frac{47}{32}\right)^{2}$$

Square \frac{47}{32} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{47}{16}t+\frac{2209}{1024}=-\frac{5}{4}+\frac{2209}{1024}$$

Add -\frac{5}{4} to \frac{2209}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+\frac{47}{16}t+\frac{2209}{1024}=\frac{929}{1024}$$

Factor t^{2}+\frac{47}{16}t+\frac{2209}{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}.

$$\left(t+\frac{47}{32}\right)^{2}=\frac{929}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{47}{32}\right)^{2}}=\sqrt{\frac{929}{1024}}$$

Simplify.

$$t+\frac{47}{32}=\frac{\sqrt{929}}{32} t+\frac{47}{32}=-\frac{\sqrt{929}}{32}$$

Subtract \frac{47}{32} from both sides of the equation.

$$t=\frac{\sqrt{929}-47}{32} t=\frac{-\sqrt{929}-47}{32}$$