Solve f(t)=3t-7t^2

$$f ( t ) = 3 t - 7 t ^ { 2 }$$

Factor

$$t\left(3-7t\right)$$

Solution Steps

Factor out t.

$$t\left(3-7t\right)$$

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.

$$-7t^{2}+3t=0$$

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.

$$t=\frac{-3±\sqrt{3^{2}}}{2\left(-7\right)}$$

Take the square root of 3^{2}.

$$t=\frac{-3±3}{2\left(-7\right)}$$

Multiply 2 times -7.

$$t=\frac{-3±3}{-14}$$

Now solve the equation t=\frac{-3±3}{-14} when ± is plus. Add -3 to 3.

$$t=\frac{0}{-14}$$

Divide 0 by -14.

$$t=0$$

Now solve the equation t=\frac{-3±3}{-14} when ± is minus. Subtract 3 from -3.

$$t=-\frac{6}{-14}$$

Reduce the fraction \frac{-6}{-14} to lowest terms by extracting and canceling out 2.

$$t=\frac{3}{7}$$

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 \frac{3}{7} for x_{2}.

$$-7t^{2}+3t=-7t\left(t-\frac{3}{7}\right)$$

Subtract \frac{3}{7} from t by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-7t^{2}+3t=-7t\times \frac{-7t+3}{-7}$$

Cancel out 7, the greatest common factor in -7 and -7.

$$-7t^{2}+3t=t\left(-7t+3\right)$$

Evaluate

$$t\left(3-7t\right)$$