Solve 16z^2-40z-25

$$16 z ^ { 2 } - 40 z - 25$$

Factor

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

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.

$$16z^{2}-40z-25=0$$

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.

$$z=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 16\left(-25\right)}}{2\times 16}$$

Square -40.

$$z=\frac{-\left(-40\right)±\sqrt{1600-4\times 16\left(-25\right)}}{2\times 16}$$

Multiply -4 times 16.

$$z=\frac{-\left(-40\right)±\sqrt{1600-64\left(-25\right)}}{2\times 16}$$

Multiply -64 times -25.

$$z=\frac{-\left(-40\right)±\sqrt{1600+1600}}{2\times 16}$$

Add 1600 to 1600.

$$z=\frac{-\left(-40\right)±\sqrt{3200}}{2\times 16}$$

Take the square root of 3200.

$$z=\frac{-\left(-40\right)±40\sqrt{2}}{2\times 16}$$

The opposite of -40 is 40.

$$z=\frac{40±40\sqrt{2}}{2\times 16}$$

Multiply 2 times 16.

$$z=\frac{40±40\sqrt{2}}{32}$$

Now solve the equation z=\frac{40±40\sqrt{2}}{32} when ± is plus. Add 40 to 40\sqrt{2}.

$$z=\frac{40\sqrt{2}+40}{32}$$

Divide 40+40\sqrt{2} by 32.

$$z=\frac{5\sqrt{2}+5}{4}$$

Now solve the equation z=\frac{40±40\sqrt{2}}{32} when ± is minus. Subtract 40\sqrt{2} from 40.

$$z=\frac{40-40\sqrt{2}}{32}$$

Divide 40-40\sqrt{2} by 32.

$$z=\frac{5-5\sqrt{2}}{4}$$

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

$$16z^{2}-40z-25=16\left(z-\frac{5\sqrt{2}+5}{4}\right)\left(z-\frac{5-5\sqrt{2}}{4}\right)$$

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

$$x ^ 2 -\frac{5}{2}x -\frac{25}{16} = 0$$

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

$$r + s = \frac{5}{2} rs = -\frac{25}{16}$$

Two numbers r and s sum up to \frac{5}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{5}{2} = \frac{5}{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>

$$r = \frac{5}{4} - u s = \frac{5}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{25}{16}

$$(\frac{5}{4} - u) (\frac{5}{4} + u) = -\frac{25}{16}$$

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

$$\frac{25}{16} - u^2 = -\frac{25}{16}$$

Simplify the expression by subtracting \frac{25}{16} on both sides

$$-u^2 = -\frac{25}{16}-\frac{25}{16} = -\frac{25}{8}$$

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

$$u^2 = \frac{25}{8} u = \pm\sqrt{\frac{25}{8}} = \pm \frac{5}{\sqrt{8}}$$

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

$$r =\frac{5}{4} - \frac{5}{\sqrt{8}} = -0.518 s = \frac{5}{4} + \frac{5}{\sqrt{8}} = 3.018$$

Evaluate

$$16z^{2}-40z-25$$