Solve 16x^2+32x-271

$$16 { x }^{ 2 } +32x-271$$

Factor

$$16\left(x-\left(-\frac{\sqrt{287}}{4}-1\right)\right)\left(x-\left(\frac{\sqrt{287}}{4}-1\right)\right)$$

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.

$$16x^{2}+32x-271=0$$

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.

$$x=\frac{-32±\sqrt{32^{2}-4\times 16\left(-271\right)}}{2\times 16}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\times 16\left(-271\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-32±\sqrt{1024-64\left(-271\right)}}{2\times 16}$$

Multiply -64 times -271.

$$x=\frac{-32±\sqrt{1024+17344}}{2\times 16}$$

Add 1024 to 17344.

$$x=\frac{-32±\sqrt{18368}}{2\times 16}$$

Take the square root of 18368.

$$x=\frac{-32±8\sqrt{287}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-32±8\sqrt{287}}{32}$$

Now solve the equation x=\frac{-32±8\sqrt{287}}{32} when ± is plus. Add -32 to 8\sqrt{287}.

$$x=\frac{8\sqrt{287}-32}{32}$$

Divide -32+8\sqrt{287} by 32.

$$x=\frac{\sqrt{287}}{4}-1$$

Now solve the equation x=\frac{-32±8\sqrt{287}}{32} when ± is minus. Subtract 8\sqrt{287} from -32.

$$x=\frac{-8\sqrt{287}-32}{32}$$

Divide -32-8\sqrt{287} by 32.

$$x=-\frac{\sqrt{287}}{4}-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1+\frac{\sqrt{287}}{4} for x_{1} and -1-\frac{\sqrt{287}}{4} for x_{2}.

$$16x^{2}+32x-271=16\left(x-\left(\frac{\sqrt{287}}{4}-1\right)\right)\left(x-\left(-\frac{\sqrt{287}}{4}-1\right)\right)$$

Evaluate

$$16x^{2}+32x-271$$