Solve 3x-6x^2-6

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

Factor

$$3\left(-2x^{2}+x-2\right)$$

Solution Steps

Factor out 3. Polynomial x-2x^{2}-2 is not factored since it does not have any rational roots.

$$3\left(x-2x^{2}-2\right)$$

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.

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

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.

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

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-6\right)\left(-6\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-3±\sqrt{9+24\left(-6\right)}}{2\left(-6\right)}$$

Multiply 24 times -6.

$$x=\frac{-3±\sqrt{9-144}}{2\left(-6\right)}$$

Add 9 to -144.

$$x=\frac{-3±\sqrt{-135}}{2\left(-6\right)}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Quadratic polynomial cannot be factored.

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

Evaluate

$$3\left(-2x^{2}+x-2\right)$$