Solve 16x^2-24x+3

$$16 { x }^{ 2 } -24x+3$$

Factor

$$16\left(x-\frac{3-\sqrt{6}}{4}\right)\left(x-\frac{\sqrt{6}+3}{4}\right)$$

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.

$$16x^{2}-24x+3=0$$

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.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 16\times 3}}{2\times 16}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 16\times 3}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-24\right)±\sqrt{576-64\times 3}}{2\times 16}$$

Multiply -64 times 3.

$$x=\frac{-\left(-24\right)±\sqrt{576-192}}{2\times 16}$$

Add 576 to -192.

$$x=\frac{-\left(-24\right)±\sqrt{384}}{2\times 16}$$

Take the square root of 384.

$$x=\frac{-\left(-24\right)±8\sqrt{6}}{2\times 16}$$

The opposite of -24 is 24.

$$x=\frac{24±8\sqrt{6}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{24±8\sqrt{6}}{32}$$

Now solve the equation x=\frac{24±8\sqrt{6}}{32} when ± is plus. Add 24 to 8\sqrt{6}.

$$x=\frac{8\sqrt{6}+24}{32}$$

Divide 24+8\sqrt{6} by 32.

$$x=\frac{\sqrt{6}+3}{4}$$

Now solve the equation x=\frac{24±8\sqrt{6}}{32} when ± is minus. Subtract 8\sqrt{6} from 24.

$$x=\frac{24-8\sqrt{6}}{32}$$

Divide 24-8\sqrt{6} by 32.

$$x=\frac{3-\sqrt{6}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+\sqrt{6}}{4} for x_{1} and \frac{3-\sqrt{6}}{4} for x_{2}.

$$16x^{2}-24x+3=16\left(x-\frac{\sqrt{6}+3}{4}\right)\left(x-\frac{3-\sqrt{6}}{4}\right)$$

Evaluate

$$16x^{2}-24x+3$$