Solve f(x)=x^2-18x-4

$$f ( x ) = x ^ { 2 } - 18 x - 4$$

Factor

$$\left(x-\left(9-\sqrt{85}\right)\right)\left(x-\left(\sqrt{85}+9\right)\right)$$

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.

$$x^{2}-18x-4=0$$

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.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-4\right)}}{2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\left(-4\right)}}{2}$$

Multiply -4 times -4.

$$x=\frac{-\left(-18\right)±\sqrt{324+16}}{2}$$

Add 324 to 16.

$$x=\frac{-\left(-18\right)±\sqrt{340}}{2}$$

Take the square root of 340.

$$x=\frac{-\left(-18\right)±2\sqrt{85}}{2}$$

The opposite of -18 is 18.

$$x=\frac{18±2\sqrt{85}}{2}$$

Now solve the equation x=\frac{18±2\sqrt{85}}{2} when ± is plus. Add 18 to 2\sqrt{85}.

$$x=\frac{2\sqrt{85}+18}{2}$$

Divide 18+2\sqrt{85} by 2.

$$x=\sqrt{85}+9$$

Now solve the equation x=\frac{18±2\sqrt{85}}{2} when ± is minus. Subtract 2\sqrt{85} from 18.

$$x=\frac{18-2\sqrt{85}}{2}$$

Divide 18-2\sqrt{85} by 2.

$$x=9-\sqrt{85}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 9+\sqrt{85} for x_{1} and 9-\sqrt{85} for x_{2}.

$$x^{2}-18x-4=\left(x-\left(\sqrt{85}+9\right)\right)\left(x-\left(9-\sqrt{85}\right)\right)$$

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.

$$x ^ 2 -18x -4 = 0$$

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

$$r + s = 18 rs = -4$$

Two numbers r and s sum up to 18 exactly when the average of the two numbers is \frac{1}{2}*18 = 9. 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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = 9 - u s = 9 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -4

$$(9 - u) (9 + u) = -4$$

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

$$81 - u^2 = -4$$

Simplify the expression by subtracting 81 on both sides

$$-u^2 = -4-81 = -85$$

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

$$u^2 = 85 u = \pm\sqrt{85} = \pm \sqrt{85}$$

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

$$r =9 - \sqrt{85} = -0.220 s = 9 + \sqrt{85} = 18.220$$

Evaluate

$$x^{2}-18x-4$$