Solve 14t^2+21t

$$14 t ^ { 2 } + 21 t$$

Factor

$$7t\left(2t+3\right)$$

Solution Steps

Factor out 7.

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

Consider 2t^{2}+3t. Factor out t.

$$t\left(2t+3\right)$$

Rewrite the complete factored expression.

$$7t\left(2t+3\right)$$

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.

$$14t^{2}+21t=0$$

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.

$$t=\frac{-21±\sqrt{21^{2}}}{2\times 14}$$

Take the square root of 21^{2}.

$$t=\frac{-21±21}{2\times 14}$$

Multiply 2 times 14.

$$t=\frac{-21±21}{28}$$

Now solve the equation t=\frac{-21±21}{28} when ± is plus. Add -21 to 21.

$$t=\frac{0}{28}$$

Divide 0 by 28.

$$t=0$$

Now solve the equation t=\frac{-21±21}{28} when ± is minus. Subtract 21 from -21.

$$t=-\frac{42}{28}$$

Reduce the fraction \frac{-42}{28} to lowest terms by extracting and canceling out 14.

$$t=-\frac{3}{2}$$

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{3}{2} for x_{2}.

$$14t^{2}+21t=14t\left(t-\left(-\frac{3}{2}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$14t^{2}+21t=14t\left(t+\frac{3}{2}\right)$$

Add \frac{3}{2} to t by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$14t^{2}+21t=14t\times \frac{2t+3}{2}$$

Cancel out 2, the greatest common factor in 14 and 2.

$$14t^{2}+21t=7t\left(2t+3\right)$$

Evaluate

$$7t\left(2t+3\right)$$