Solve 6=-16t^2+14t+4

$$6 = - 16 t ^ { 2 } + 14 t + 4$$

Solve for t

$$t=\frac{\sqrt{17}+7}{16}\approx 0.695194102$$

$$t=\frac{7-\sqrt{17}}{16}\approx 0.179805898$$

Steps Using the Quadratic Formula

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

$$-16t^{2}+14t+4=6$$

Subtract 6 from both sides.

$$-16t^{2}+14t+4-6=0$$

Subtract 6 from 4 to get -2.

$$-16t^{2}+14t-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 14 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-14±\sqrt{14^{2}-4\left(-16\right)\left(-2\right)}}{2\left(-16\right)}$$

Square 14.

$$t=\frac{-14±\sqrt{196-4\left(-16\right)\left(-2\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-14±\sqrt{196+64\left(-2\right)}}{2\left(-16\right)}$$

Multiply 64 times -2.

$$t=\frac{-14±\sqrt{196-128}}{2\left(-16\right)}$$

Add 196 to -128.

$$t=\frac{-14±\sqrt{68}}{2\left(-16\right)}$$

Take the square root of 68.

$$t=\frac{-14±2\sqrt{17}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-14±2\sqrt{17}}{-32}$$

Now solve the equation t=\frac{-14±2\sqrt{17}}{-32} when ± is plus. Add -14 to 2\sqrt{17}.

$$t=\frac{2\sqrt{17}-14}{-32}$$

Divide -14+2\sqrt{17} by -32.

$$t=\frac{7-\sqrt{17}}{16}$$

Now solve the equation t=\frac{-14±2\sqrt{17}}{-32} when ± is minus. Subtract 2\sqrt{17} from -14.

$$t=\frac{-2\sqrt{17}-14}{-32}$$

Divide -14-2\sqrt{17} by -32.

$$t=\frac{\sqrt{17}+7}{16}$$

The equation is now solved.

$$t=\frac{7-\sqrt{17}}{16} t=\frac{\sqrt{17}+7}{16}$$

Steps for Completing the Square

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

$$-16t^{2}+14t+4=6$$

Subtract 4 from both sides.

$$-16t^{2}+14t=6-4$$

Subtract 4 from 6 to get 2.

$$-16t^{2}+14t=2$$

Divide both sides by -16.

$$\frac{-16t^{2}+14t}{-16}=\frac{2}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$t^{2}+\frac{14}{-16}t=\frac{2}{-16}$$

Reduce the fraction \frac{14}{-16} to lowest terms by extracting and canceling out 2.

$$t^{2}-\frac{7}{8}t=\frac{2}{-16}$$

Reduce the fraction \frac{2}{-16} to lowest terms by extracting and canceling out 2.

$$t^{2}-\frac{7}{8}t=-\frac{1}{8}$$

Divide -\frac{7}{8}, the coefficient of the x term, by 2 to get -\frac{7}{16}. Then add the square of -\frac{7}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{7}{8}t+\left(-\frac{7}{16}\right)^{2}=-\frac{1}{8}+\left(-\frac{7}{16}\right)^{2}$$

Square -\frac{7}{16} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{7}{8}t+\frac{49}{256}=-\frac{1}{8}+\frac{49}{256}$$

Add -\frac{1}{8} to \frac{49}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{7}{8}t+\frac{49}{256}=\frac{17}{256}$$

Factor t^{2}-\frac{7}{8}t+\frac{49}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{7}{16}\right)^{2}=\frac{17}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{7}{16}\right)^{2}}=\sqrt{\frac{17}{256}}$$

Simplify.

$$t-\frac{7}{16}=\frac{\sqrt{17}}{16} t-\frac{7}{16}=-\frac{\sqrt{17}}{16}$$

Add \frac{7}{16} to both sides of the equation.

$$t=\frac{\sqrt{17}+7}{16} t=\frac{7-\sqrt{17}}{16}$$