Solve 18+21d-8d^2

$$18 + 21 d - 8 d ^ { 2 }$$

Factor

$$-8\left(d-\frac{21-3\sqrt{113}}{16}\right)\left(d-\frac{3\sqrt{113}+21}{16}\right)$$

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.

$$-8d^{2}+21d+18=0$$

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.

$$d=\frac{-21±\sqrt{21^{2}-4\left(-8\right)\times 18}}{2\left(-8\right)}$$

Square 21.

$$d=\frac{-21±\sqrt{441-4\left(-8\right)\times 18}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$d=\frac{-21±\sqrt{441+32\times 18}}{2\left(-8\right)}$$

Multiply 32 times 18.

$$d=\frac{-21±\sqrt{441+576}}{2\left(-8\right)}$$

Add 441 to 576.

$$d=\frac{-21±\sqrt{1017}}{2\left(-8\right)}$$

Take the square root of 1017.

$$d=\frac{-21±3\sqrt{113}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$d=\frac{-21±3\sqrt{113}}{-16}$$

Now solve the equation d=\frac{-21±3\sqrt{113}}{-16} when ± is plus. Add -21 to 3\sqrt{113}.

$$d=\frac{3\sqrt{113}-21}{-16}$$

Divide -21+3\sqrt{113} by -16.

$$d=\frac{21-3\sqrt{113}}{16}$$

Now solve the equation d=\frac{-21±3\sqrt{113}}{-16} when ± is minus. Subtract 3\sqrt{113} from -21.

$$d=\frac{-3\sqrt{113}-21}{-16}$$

Divide -21-3\sqrt{113} by -16.

$$d=\frac{3\sqrt{113}+21}{16}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{21-3\sqrt{113}}{16} for x_{1} and \frac{21+3\sqrt{113}}{16} for x_{2}.

$$-8d^{2}+21d+18=-8\left(d-\frac{21-3\sqrt{113}}{16}\right)\left(d-\frac{3\sqrt{113}+21}{16}\right)$$

Evaluate

$$18+21d-8d^{2}$$