Solve a^2-a-18

$$a ^ { 2 } - a - 18$$

Factor

$$\left(a-\frac{1-\sqrt{73}}{2}\right)\left(a-\frac{\sqrt{73}+1}{2}\right)$$

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.

$$a^{2}-a-18=0$$

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.

$$a=\frac{-\left(-1\right)±\sqrt{1-4\left(-18\right)}}{2}$$

Multiply -4 times -18.

$$a=\frac{-\left(-1\right)±\sqrt{1+72}}{2}$$

Add 1 to 72.

$$a=\frac{-\left(-1\right)±\sqrt{73}}{2}$$

The opposite of -1 is 1.

$$a=\frac{1±\sqrt{73}}{2}$$

Now solve the equation a=\frac{1±\sqrt{73}}{2} when ± is plus. Add 1 to \sqrt{73}.

$$a=\frac{\sqrt{73}+1}{2}$$

Now solve the equation a=\frac{1±\sqrt{73}}{2} when ± is minus. Subtract \sqrt{73} from 1.

$$a=\frac{1-\sqrt{73}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{73}}{2} for x_{1} and \frac{1-\sqrt{73}}{2} for x_{2}.

$$a^{2}-a-18=\left(a-\frac{\sqrt{73}+1}{2}\right)\left(a-\frac{1-\sqrt{73}}{2}\right)$$

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.

$$x ^ 2 -1x -18 = 0$$

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

$$r + s = 1 rs = -18$$

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

$$r = \frac{1}{2} - u s = \frac{1}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -18

$$(\frac{1}{2} - u) (\frac{1}{2} + u) = -18$$

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

$$\frac{1}{4} - u^2 = -18$$

Simplify the expression by subtracting \frac{1}{4} on both sides

$$-u^2 = -18-\frac{1}{4} = -\frac{73}{4}$$

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

$$u^2 = \frac{73}{4} u = \pm\sqrt{\frac{73}{4}} = \pm \frac{\sqrt{73}}{2}$$

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

$$r =\frac{1}{2} - \frac{\sqrt{73}}{2} = -3.772 s = \frac{1}{2} + \frac{\sqrt{73}}{2} = 4.772$$

Evaluate

$$a^{2}-a-18$$