Solve -12x^2+6x-14

$$-12 { x }^{ 2 } +6x-14$$

Factor

$$2\left(-6x^{2}+3x-7\right)$$

Solution Steps

Factor out 2. Polynomial -6x^{2}+3x-7 is not factored since it does not have any rational roots.

$$2\left(-6x^{2}+3x-7\right)$$

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.

$$-12x^{2}+6x-14=0$$

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.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-12\right)\left(-14\right)}}{2\left(-12\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-12\right)\left(-14\right)}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-6±\sqrt{36+48\left(-14\right)}}{2\left(-12\right)}$$

Multiply 48 times -14.

$$x=\frac{-6±\sqrt{36-672}}{2\left(-12\right)}$$

Add 36 to -672.

$$x=\frac{-6±\sqrt{-636}}{2\left(-12\right)}$$

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

$$-12x^{2}+6x-14$$

Evaluate

$$-12x^{2}+6x-14$$