Solve 2z^2-16z-35

$$2 z ^ { 2 } - 16 z - 35$$

Factor

$$2\left(z-\left(-\frac{\sqrt{134}}{2}+4\right)\right)\left(z-\left(\frac{\sqrt{134}}{2}+4\right)\right)$$

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.

$$2z^{2}-16z-35=0$$

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.

$$z=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 2\left(-35\right)}}{2\times 2}$$

Square -16.

$$z=\frac{-\left(-16\right)±\sqrt{256-4\times 2\left(-35\right)}}{2\times 2}$$

Multiply -4 times 2.

$$z=\frac{-\left(-16\right)±\sqrt{256-8\left(-35\right)}}{2\times 2}$$

Multiply -8 times -35.

$$z=\frac{-\left(-16\right)±\sqrt{256+280}}{2\times 2}$$

Add 256 to 280.

$$z=\frac{-\left(-16\right)±\sqrt{536}}{2\times 2}$$

Take the square root of 536.

$$z=\frac{-\left(-16\right)±2\sqrt{134}}{2\times 2}$$

The opposite of -16 is 16.

$$z=\frac{16±2\sqrt{134}}{2\times 2}$$

Multiply 2 times 2.

$$z=\frac{16±2\sqrt{134}}{4}$$

Now solve the equation z=\frac{16±2\sqrt{134}}{4} when ± is plus. Add 16 to 2\sqrt{134}.

$$z=\frac{2\sqrt{134}+16}{4}$$

Divide 16+2\sqrt{134} by 4.

$$z=\frac{\sqrt{134}}{2}+4$$

Now solve the equation z=\frac{16±2\sqrt{134}}{4} when ± is minus. Subtract 2\sqrt{134} from 16.

$$z=\frac{16-2\sqrt{134}}{4}$$

Divide 16-2\sqrt{134} by 4.

$$z=-\frac{\sqrt{134}}{2}+4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4+\frac{\sqrt{134}}{2} for x_{1} and 4-\frac{\sqrt{134}}{2} for x_{2}.

$$2z^{2}-16z-35=2\left(z-\left(\frac{\sqrt{134}}{2}+4\right)\right)\left(z-\left(-\frac{\sqrt{134}}{2}+4\right)\right)$$

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 2

$$x ^ 2 -8x -\frac{35}{2} = 0$$

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

$$r + s = 8 rs = -\frac{35}{2}$$

Two numbers r and s sum up to 8 exactly when the average of the two numbers is \frac{1}{2}*8 = 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>

$$r = 4 - u s = 4 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{35}{2}

$$(4 - u) (4 + u) = -\frac{35}{2}$$

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

$$16 - u^2 = -\frac{35}{2}$$

Simplify the expression by subtracting 16 on both sides

$$-u^2 = -\frac{35}{2}-16 = -\frac{67}{2}$$

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

$$u^2 = \frac{67}{2} u = \pm\sqrt{\frac{67}{2}} = \pm \frac{\sqrt{67}}{\sqrt{2}}$$

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

$$r =4 - \frac{\sqrt{67}}{\sqrt{2}} = -1.788 s = 4 + \frac{\sqrt{67}}{\sqrt{2}} = 9.788$$

Evaluate

$$2z^{2}-16z-35$$