Solve 6d^2-21d

$$6 d ^ { 2 } - 21 d$$

Factor

$$3d\left(2d-7\right)$$

Solution Steps

Factor out 3.

$$3\left(2d^{2}-7d\right)$$

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

$$d\left(2d-7\right)$$

Rewrite the complete factored expression.

$$3d\left(2d-7\right)$$

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.

$$6d^{2}-21d=0$$

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.

$$d=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}}}{2\times 6}$$

Take the square root of \left(-21\right)^{2}.

$$d=\frac{-\left(-21\right)±21}{2\times 6}$$

The opposite of -21 is 21.

$$d=\frac{21±21}{2\times 6}$$

Multiply 2 times 6.

$$d=\frac{21±21}{12}$$

Now solve the equation d=\frac{21±21}{12} when ± is plus. Add 21 to 21.

$$d=\frac{42}{12}$$

Reduce the fraction \frac{42}{12} to lowest terms by extracting and canceling out 6.

$$d=\frac{7}{2}$$

Now solve the equation d=\frac{21±21}{12} when ± is minus. Subtract 21 from 21.

$$d=\frac{0}{12}$$

Divide 0 by 12.

$$d=0$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7}{2} for x_{1} and 0 for x_{2}.

$$6d^{2}-21d=6\left(d-\frac{7}{2}\right)d$$

Subtract \frac{7}{2} from d by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$6d^{2}-21d=6\times \frac{2d-7}{2}d$$

Cancel out 2, the greatest common factor in 6 and 2.

$$6d^{2}-21d=3\left(2d-7\right)d$$

Evaluate

$$3d\left(2d-7\right)$$