Solve 2x^2-4x+1-6x^2+5

$$2 { x }^{ 2 } -4x+1-6 { x }^{ 2 } +5$$

Evaluate

$$6-4x-4x^{2}$$

Solution Steps

Combine 2x^{2} and -6x^{2} to get -4x^{2}.

$$-4x^{2}-4x+1+5$$

Add 1 and 5 to get 6.

$$-4x^{2}-4x+6$$

Factor

$$-4\left(x-\frac{-\sqrt{7}-1}{2}\right)\left(x-\frac{\sqrt{7}-1}{2}\right)$$

Steps Using the Quadratic Formula

Combine 2x^{2} and -6x^{2} to get -4x^{2}.

$$factor(-4x^{2}-4x+1+5)$$

Add 1 and 5 to get 6.

$$factor(-4x^{2}-4x+6)$$

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.

$$-4x^{2}-4x+6=0$$

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.

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

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-4\right)\times 6}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-4\right)±\sqrt{16+16\times 6}}{2\left(-4\right)}$$

Multiply 16 times 6.

$$x=\frac{-\left(-4\right)±\sqrt{16+96}}{2\left(-4\right)}$$

Add 16 to 96.

$$x=\frac{-\left(-4\right)±\sqrt{112}}{2\left(-4\right)}$$

Take the square root of 112.

$$x=\frac{-\left(-4\right)±4\sqrt{7}}{2\left(-4\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{7}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{4±4\sqrt{7}}{-8}$$

Now solve the equation x=\frac{4±4\sqrt{7}}{-8} when ± is plus. Add 4 to 4\sqrt{7}.

$$x=\frac{4\sqrt{7}+4}{-8}$$

Divide 4+4\sqrt{7} by -8.

$$x=\frac{-\sqrt{7}-1}{2}$$

Now solve the equation x=\frac{4±4\sqrt{7}}{-8} when ± is minus. Subtract 4\sqrt{7} from 4.

$$x=\frac{4-4\sqrt{7}}{-8}$$

Divide 4-4\sqrt{7} by -8.

$$x=\frac{\sqrt{7}-1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1-\sqrt{7}}{2} for x_{1} and \frac{-1+\sqrt{7}}{2} for x_{2}.

$$-4x^{2}-4x+6=-4\left(x-\frac{-\sqrt{7}-1}{2}\right)\left(x-\frac{\sqrt{7}-1}{2}\right)$$