Solve 84t-12t^2

$$84 t - 12 t ^ { 2 }$$

Factor

$$12t\left(7-t\right)$$

Solution Steps

Factor out 12.

$$12\left(7t-t^{2}\right)$$

Consider 7t-t^{2}. Factor out t.

$$t\left(7-t\right)$$

Rewrite the complete factored expression.

$$12t\left(-t+7\right)$$

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.

$$-12t^{2}+84t=0$$

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.

$$t=\frac{-84±\sqrt{84^{2}}}{2\left(-12\right)}$$

Take the square root of 84^{2}.

$$t=\frac{-84±84}{2\left(-12\right)}$$

Multiply 2 times -12.

$$t=\frac{-84±84}{-24}$$

Now solve the equation t=\frac{-84±84}{-24} when ± is plus. Add -84 to 84.

$$t=\frac{0}{-24}$$

Divide 0 by -24.

$$t=0$$

Now solve the equation t=\frac{-84±84}{-24} when ± is minus. Subtract 84 from -84.

$$t=-\frac{168}{-24}$$

Divide -168 by -24.

$$t=7$$

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 7 for x_{2}.

$$-12t^{2}+84t=-12t\left(t-7\right)$$

Evaluate

$$12t\left(7-t\right)$$