Solve f(x)=4x+10-16x^2

$$f ( x ) = 4 x + 10 - 16 x ^ { 2 }$$

Factor

$$-16\left(x-\frac{1-\sqrt{41}}{8}\right)\left(x-\frac{\sqrt{41}+1}{8}\right)$$

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.

$$-16x^{2}+4x+10=0$$

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.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-16\right)\times 10}}{2\left(-16\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-16\right)\times 10}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-4±\sqrt{16+64\times 10}}{2\left(-16\right)}$$

Multiply 64 times 10.

$$x=\frac{-4±\sqrt{16+640}}{2\left(-16\right)}$$

Add 16 to 640.

$$x=\frac{-4±\sqrt{656}}{2\left(-16\right)}$$

Take the square root of 656.

$$x=\frac{-4±4\sqrt{41}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-4±4\sqrt{41}}{-32}$$

Now solve the equation x=\frac{-4±4\sqrt{41}}{-32} when ± is plus. Add -4 to 4\sqrt{41}.

$$x=\frac{4\sqrt{41}-4}{-32}$$

Divide -4+4\sqrt{41} by -32.

$$x=\frac{1-\sqrt{41}}{8}$$

Now solve the equation x=\frac{-4±4\sqrt{41}}{-32} when ± is minus. Subtract 4\sqrt{41} from -4.

$$x=\frac{-4\sqrt{41}-4}{-32}$$

Divide -4-4\sqrt{41} by -32.

$$x=\frac{\sqrt{41}+1}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1-\sqrt{41}}{8} for x_{1} and \frac{1+\sqrt{41}}{8} for x_{2}.

$$-16x^{2}+4x+10=-16\left(x-\frac{1-\sqrt{41}}{8}\right)\left(x-\frac{\sqrt{41}+1}{8}\right)$$

Evaluate

$$10+4x-16x^{2}$$