Solve {l}{6a^2+12a}{-6}

$$\left. \begin{array} { l } { 6 a ^ { 2 } + 12 a } \\ { - 6 } \end{array} \right.$$

Factor

$$6\left(a-\left(-\sqrt{2}-1\right)\right)\left(a-\left(\sqrt{2}-1\right)\right)$$

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.

$$6a^{2}+12a-6=0$$

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.

$$a=\frac{-12±\sqrt{12^{2}-4\times 6\left(-6\right)}}{2\times 6}$$

Square 12.

$$a=\frac{-12±\sqrt{144-4\times 6\left(-6\right)}}{2\times 6}$$

Multiply -4 times 6.

$$a=\frac{-12±\sqrt{144-24\left(-6\right)}}{2\times 6}$$

Multiply -24 times -6.

$$a=\frac{-12±\sqrt{144+144}}{2\times 6}$$

Add 144 to 144.

$$a=\frac{-12±\sqrt{288}}{2\times 6}$$

Take the square root of 288.

$$a=\frac{-12±12\sqrt{2}}{2\times 6}$$

Multiply 2 times 6.

$$a=\frac{-12±12\sqrt{2}}{12}$$

Now solve the equation a=\frac{-12±12\sqrt{2}}{12} when ± is plus. Add -12 to 12\sqrt{2}.

$$a=\frac{12\sqrt{2}-12}{12}$$

Divide -12+12\sqrt{2} by 12.

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

Now solve the equation a=\frac{-12±12\sqrt{2}}{12} when ± is minus. Subtract 12\sqrt{2} from -12.

$$a=\frac{-12\sqrt{2}-12}{12}$$

Divide -12-12\sqrt{2} by 12.

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

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

$$6a^{2}+12a-6=6\left(a-\left(\sqrt{2}-1\right)\right)\left(a-\left(-\sqrt{2}-1\right)\right)$$

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 6

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

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

$$r + s = -2 rs = -1$$

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 = -1

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

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

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

Simplify the expression by subtracting 1 on both sides

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

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

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

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

$$r =-1 - \sqrt{2} = -2.414 s = -1 + \sqrt{2} = 0.414$$

Evaluate

$$6\left(a^{2}+2a-1\right)$$