Solve 18x^2+32x-16

$$18 { x }^{ 2 } +32x-16$$

Factor

$$18\left(x-\frac{-2\sqrt{34}-8}{9}\right)\left(x-\frac{2\sqrt{34}-8}{9}\right)$$

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.

$$18x^{2}+32x-16=0$$

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.

$$x=\frac{-32±\sqrt{32^{2}-4\times 18\left(-16\right)}}{2\times 18}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\times 18\left(-16\right)}}{2\times 18}$$

Multiply -4 times 18.

$$x=\frac{-32±\sqrt{1024-72\left(-16\right)}}{2\times 18}$$

Multiply -72 times -16.

$$x=\frac{-32±\sqrt{1024+1152}}{2\times 18}$$

Add 1024 to 1152.

$$x=\frac{-32±\sqrt{2176}}{2\times 18}$$

Take the square root of 2176.

$$x=\frac{-32±8\sqrt{34}}{2\times 18}$$

Multiply 2 times 18.

$$x=\frac{-32±8\sqrt{34}}{36}$$

Now solve the equation x=\frac{-32±8\sqrt{34}}{36} when ± is plus. Add -32 to 8\sqrt{34}.

$$x=\frac{8\sqrt{34}-32}{36}$$

Divide -32+8\sqrt{34} by 36.

$$x=\frac{2\sqrt{34}-8}{9}$$

Now solve the equation x=\frac{-32±8\sqrt{34}}{36} when ± is minus. Subtract 8\sqrt{34} from -32.

$$x=\frac{-8\sqrt{34}-32}{36}$$

Divide -32-8\sqrt{34} by 36.

$$x=\frac{-2\sqrt{34}-8}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-8+2\sqrt{34}}{9} for x_{1} and \frac{-8-2\sqrt{34}}{9} for x_{2}.

$$18x^{2}+32x-16=18\left(x-\frac{2\sqrt{34}-8}{9}\right)\left(x-\frac{-2\sqrt{34}-8}{9}\right)$$

Evaluate

$$18x^{2}+32x-16$$