Solve -3x^2-6x+12

$$- 3 x ^ { 2 } - 6 x + 12$$

Factor

$$-3\left(x-\left(-\sqrt{5}-1\right)\right)\left(x-\left(\sqrt{5}-1\right)\right)$$

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.

$$-3x^{2}-6x+12=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-3\right)\times 12}}{2\left(-3\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-3\right)\times 12}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-6\right)±\sqrt{36+12\times 12}}{2\left(-3\right)}$$

Multiply 12 times 12.

$$x=\frac{-\left(-6\right)±\sqrt{36+144}}{2\left(-3\right)}$$

Add 36 to 144.

$$x=\frac{-\left(-6\right)±\sqrt{180}}{2\left(-3\right)}$$

Take the square root of 180.

$$x=\frac{-\left(-6\right)±6\sqrt{5}}{2\left(-3\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±6\sqrt{5}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{6±6\sqrt{5}}{-6}$$

Now solve the equation x=\frac{6±6\sqrt{5}}{-6} when ± is plus. Add 6 to 6\sqrt{5}.

$$x=\frac{6\sqrt{5}+6}{-6}$$

Divide 6+6\sqrt{5} by -6.

$$x=-\left(\sqrt{5}+1\right)$$

Now solve the equation x=\frac{6±6\sqrt{5}}{-6} when ± is minus. Subtract 6\sqrt{5} from 6.

$$x=\frac{6-6\sqrt{5}}{-6}$$

Divide 6-6\sqrt{5} by -6.

$$x=\sqrt{5}-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\left(1+\sqrt{5}\right) for x_{1} and -1+\sqrt{5} for x_{2}.

$$-3x^{2}-6x+12=-3\left(x-\left(-\left(\sqrt{5}+1\right)\right)\right)\left(x-\left(\sqrt{5}-1\right)\right)$$

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.

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

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

$$r + s = -2 rs = -4$$

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 = -4

$$(-1 - u) (-1 + u) = -4$$

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

$$1 - u^2 = -4$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -4-1 = -5$$

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

$$u^2 = 5 u = \pm\sqrt{5} = \pm \sqrt{5}$$

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

$$r =-1 - \sqrt{5} = -3.236 s = -1 + \sqrt{5} = 1.236$$

Evaluate

$$12-6x-3x^{2}$$