Solve f(t)=126t-6t^2

$$f ( t ) = 126 t - 6 t ^ { 2 }$$

Factor

$$6t\left(21-t\right)$$

Solution Steps

Factor out 6.

$$6\left(21t-t^{2}\right)$$

Consider 21t-t^{2}. Factor out t.

$$t\left(21-t\right)$$

Rewrite the complete factored expression.

$$6t\left(-t+21\right)$$

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.

$$-6t^{2}+126t=0$$

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.

$$t=\frac{-126±\sqrt{126^{2}}}{2\left(-6\right)}$$

Take the square root of 126^{2}.

$$t=\frac{-126±126}{2\left(-6\right)}$$

Multiply 2 times -6.

$$t=\frac{-126±126}{-12}$$

Now solve the equation t=\frac{-126±126}{-12} when ± is plus. Add -126 to 126.

$$t=\frac{0}{-12}$$

Divide 0 by -12.

$$t=0$$

Now solve the equation t=\frac{-126±126}{-12} when ± is minus. Subtract 126 from -126.

$$t=-\frac{252}{-12}$$

Divide -252 by -12.

$$t=21$$

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 21 for x_{2}.

$$-6t^{2}+126t=-6t\left(t-21\right)$$

Evaluate

$$6t\left(21-t\right)$$