Solve 9x^2-18x-5

$$9 x ^ { 2 } - 18 x - 5$$

Factor

$$9\left(x-\left(-\frac{\sqrt{14}}{3}+1\right)\right)\left(x-\left(\frac{\sqrt{14}}{3}+1\right)\right)$$

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.

$$9x^{2}-18x-5=0$$

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.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 9\left(-5\right)}}{2\times 9}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 9\left(-5\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-18\right)±\sqrt{324-36\left(-5\right)}}{2\times 9}$$

Multiply -36 times -5.

$$x=\frac{-\left(-18\right)±\sqrt{324+180}}{2\times 9}$$

Add 324 to 180.

$$x=\frac{-\left(-18\right)±\sqrt{504}}{2\times 9}$$

Take the square root of 504.

$$x=\frac{-\left(-18\right)±6\sqrt{14}}{2\times 9}$$

The opposite of -18 is 18.

$$x=\frac{18±6\sqrt{14}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{18±6\sqrt{14}}{18}$$

Now solve the equation x=\frac{18±6\sqrt{14}}{18} when ± is plus. Add 18 to 6\sqrt{14}.

$$x=\frac{6\sqrt{14}+18}{18}$$

Divide 18+6\sqrt{14} by 18.

$$x=\frac{\sqrt{14}}{3}+1$$

Now solve the equation x=\frac{18±6\sqrt{14}}{18} when ± is minus. Subtract 6\sqrt{14} from 18.

$$x=\frac{18-6\sqrt{14}}{18}$$

Divide 18-6\sqrt{14} by 18.

$$x=-\frac{\sqrt{14}}{3}+1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1+\frac{\sqrt{14}}{3} for x_{1} and 1-\frac{\sqrt{14}}{3} for x_{2}.

$$9x^{2}-18x-5=9\left(x-\left(\frac{\sqrt{14}}{3}+1\right)\right)\left(x-\left(-\frac{\sqrt{14}}{3}+1\right)\right)$$

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 9

$$x ^ 2 -2x -\frac{5}{9} = 0$$

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

$$r + s = 2 rs = -\frac{5}{9}$$

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

$$r = 1 - u s = 1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{9}

$$(1 - u) (1 + u) = -\frac{5}{9}$$

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

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

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -\frac{5}{9}-1 = -\frac{14}{9}$$

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

$$u^2 = \frac{14}{9} u = \pm\sqrt{\frac{14}{9}} = \pm \frac{\sqrt{14}}{3}$$

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

$$r =1 - \frac{\sqrt{14}}{3} = -0.247 s = 1 + \frac{\sqrt{14}}{3} = 2.247$$

Evaluate

$$9x^{2}-18x-5$$