Solve m^2-2m-188

$$m ^ { 2 } - 2 m - 188$$

Factor

$$\left(m-\left(1-3\sqrt{21}\right)\right)\left(m-\left(3\sqrt{21}+1\right)\right)$$

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.

$$m^{2}-2m-188=0$$

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.

$$m=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-188\right)}}{2}$$

Square -2.

$$m=\frac{-\left(-2\right)±\sqrt{4-4\left(-188\right)}}{2}$$

Multiply -4 times -188.

$$m=\frac{-\left(-2\right)±\sqrt{4+752}}{2}$$

Add 4 to 752.

$$m=\frac{-\left(-2\right)±\sqrt{756}}{2}$$

Take the square root of 756.

$$m=\frac{-\left(-2\right)±6\sqrt{21}}{2}$$

The opposite of -2 is 2.

$$m=\frac{2±6\sqrt{21}}{2}$$

Now solve the equation m=\frac{2±6\sqrt{21}}{2} when ± is plus. Add 2 to 6\sqrt{21}.

$$m=\frac{6\sqrt{21}+2}{2}$$

Divide 2+6\sqrt{21} by 2.

$$m=3\sqrt{21}+1$$

Now solve the equation m=\frac{2±6\sqrt{21}}{2} when ± is minus. Subtract 6\sqrt{21} from 2.

$$m=\frac{2-6\sqrt{21}}{2}$$

Divide 2-6\sqrt{21} by 2.

$$m=1-3\sqrt{21}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1+3\sqrt{21} for x_{1} and 1-3\sqrt{21} for x_{2}.

$$m^{2}-2m-188=\left(m-\left(3\sqrt{21}+1\right)\right)\left(m-\left(1-3\sqrt{21}\right)\right)$$

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.

$$x ^ 2 -2x -188 = 0$$

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

$$r + s = 2 rs = -188$$

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 = -188

$$(1 - u) (1 + u) = -188$$

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

$$1 - u^2 = -188$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -188-1 = -189$$

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

$$u^2 = 189 u = \pm\sqrt{189} = \pm \sqrt{189}$$

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

$$r =1 - \sqrt{189} = -12.748 s = 1 + \sqrt{189} = 14.748$$

Evaluate

$$m^{2}-2m-188$$