Solve 18+3y-4y^2

$$18 + 3 y - 4 y ^ { 2 }$$

Factor

$$-4\left(y-\frac{3-3\sqrt{33}}{8}\right)\left(y-\frac{3\sqrt{33}+3}{8}\right)$$

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.

$$-4y^{2}+3y+18=0$$

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.

$$y=\frac{-3±\sqrt{3^{2}-4\left(-4\right)\times 18}}{2\left(-4\right)}$$

Square 3.

$$y=\frac{-3±\sqrt{9-4\left(-4\right)\times 18}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$y=\frac{-3±\sqrt{9+16\times 18}}{2\left(-4\right)}$$

Multiply 16 times 18.

$$y=\frac{-3±\sqrt{9+288}}{2\left(-4\right)}$$

Add 9 to 288.

$$y=\frac{-3±\sqrt{297}}{2\left(-4\right)}$$

Take the square root of 297.

$$y=\frac{-3±3\sqrt{33}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$y=\frac{-3±3\sqrt{33}}{-8}$$

Now solve the equation y=\frac{-3±3\sqrt{33}}{-8} when ± is plus. Add -3 to 3\sqrt{33}.

$$y=\frac{3\sqrt{33}-3}{-8}$$

Divide -3+3\sqrt{33} by -8.

$$y=\frac{3-3\sqrt{33}}{8}$$

Now solve the equation y=\frac{-3±3\sqrt{33}}{-8} when ± is minus. Subtract 3\sqrt{33} from -3.

$$y=\frac{-3\sqrt{33}-3}{-8}$$

Divide -3-3\sqrt{33} by -8.

$$y=\frac{3\sqrt{33}+3}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3-3\sqrt{33}}{8} for x_{1} and \frac{3+3\sqrt{33}}{8} for x_{2}.

$$-4y^{2}+3y+18=-4\left(y-\frac{3-3\sqrt{33}}{8}\right)\left(y-\frac{3\sqrt{33}+3}{8}\right)$$

Evaluate

$$18+3y-4y^{2}$$