Solve 16z^2-24z+4

$$16 z ^ { 2 } - 24 z + 4$$

Factor

$$16\left(z-\frac{3-\sqrt{5}}{4}\right)\left(z-\frac{\sqrt{5}+3}{4}\right)$$

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.

$$16z^{2}-24z+4=0$$

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.

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

Square -24.

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

Multiply -4 times 16.

$$z=\frac{-\left(-24\right)±\sqrt{576-64\times 4}}{2\times 16}$$

Multiply -64 times 4.

$$z=\frac{-\left(-24\right)±\sqrt{576-256}}{2\times 16}$$

Add 576 to -256.

$$z=\frac{-\left(-24\right)±\sqrt{320}}{2\times 16}$$

Take the square root of 320.

$$z=\frac{-\left(-24\right)±8\sqrt{5}}{2\times 16}$$

The opposite of -24 is 24.

$$z=\frac{24±8\sqrt{5}}{2\times 16}$$

Multiply 2 times 16.

$$z=\frac{24±8\sqrt{5}}{32}$$

Now solve the equation z=\frac{24±8\sqrt{5}}{32} when ± is plus. Add 24 to 8\sqrt{5}.

$$z=\frac{8\sqrt{5}+24}{32}$$

Divide 24+8\sqrt{5} by 32.

$$z=\frac{\sqrt{5}+3}{4}$$

Now solve the equation z=\frac{24±8\sqrt{5}}{32} when ± is minus. Subtract 8\sqrt{5} from 24.

$$z=\frac{24-8\sqrt{5}}{32}$$

Divide 24-8\sqrt{5} by 32.

$$z=\frac{3-\sqrt{5}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+\sqrt{5}}{4} for x_{1} and \frac{3-\sqrt{5}}{4} for x_{2}.

$$16z^{2}-24z+4=16\left(z-\frac{\sqrt{5}+3}{4}\right)\left(z-\frac{3-\sqrt{5}}{4}\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 16

$$x ^ 2 -\frac{3}{2}x +\frac{1}{4} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = \frac{3}{2} rs = \frac{1}{4}$$

Two numbers r and s sum up to \frac{3}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{3}{2} = \frac{3}{4}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{3}{4} - u s = \frac{3}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{4}

$$(\frac{3}{4} - u) (\frac{3}{4} + u) = \frac{1}{4}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{9}{16} - u^2 = \frac{1}{4}$$

Simplify the expression by subtracting \frac{9}{16} on both sides

$$-u^2 = \frac{1}{4}-\frac{9}{16} = -\frac{5}{16}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{5}{16} u = \pm\sqrt{\frac{5}{16}} = \pm \frac{\sqrt{5}}{4}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{3}{4} - \frac{\sqrt{5}}{4} = 0.191 s = \frac{3}{4} + \frac{\sqrt{5}}{4} = 1.309$$

Evaluate

$$16z^{2}-24z+4$$