Solve 104t-16t^2

$$104t-16 { t }^{ 2 }$$

Factor

$$8t\left(13-2t\right)$$

Solution Steps

Factor out 8.

$$8\left(13t-2t^{2}\right)$$

Consider 13t-2t^{2}. Factor out t.

$$t\left(13-2t\right)$$

Rewrite the complete factored expression.

$$8t\left(-2t+13\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-16t^{2}+104t=0$$

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.

$$t=\frac{-104±\sqrt{104^{2}}}{2\left(-16\right)}$$

Take the square root of 104^{2}.

$$t=\frac{-104±104}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-104±104}{-32}$$

Now solve the equation t=\frac{-104±104}{-32} when ± is plus. Add -104 to 104.

$$t=\frac{0}{-32}$$

Divide 0 by -32.

$$t=0$$

Now solve the equation t=\frac{-104±104}{-32} when ± is minus. Subtract 104 from -104.

$$t=-\frac{208}{-32}$$

Reduce the fraction \frac{-208}{-32} to lowest terms by extracting and canceling out 16.

$$t=\frac{13}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 0 for x_{1} and \frac{13}{2} for x_{2}.

$$-16t^{2}+104t=-16t\left(t-\frac{13}{2}\right)$$

Subtract \frac{13}{2} from t by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-16t^{2}+104t=-16t\times \frac{-2t+13}{-2}$$

Cancel out 2, the greatest common factor in -16 and -2.

$$-16t^{2}+104t=8t\left(-2t+13\right)$$

Evaluate

$$8t\left(13-2t\right)$$