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

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

Evaluate

$$10+4x-4x^{2}$$

Solution Steps

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

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

Add 4 and 6 to get 10.

$$-4x^{2}+10+4x$$

Factor

$$-4\left(x-\frac{1-\sqrt{11}}{2}\right)\left(x-\frac{\sqrt{11}+1}{2}\right)$$

Steps Using the Quadratic Formula

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

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

Add 4 and 6 to get 10.

$$factor(-4x^{2}+10+4x)$$

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.

$$-4x^{2}+4x+10=0$$

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.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-4\right)\times 10}}{2\left(-4\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-4\right)\times 10}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-4±\sqrt{16+16\times 10}}{2\left(-4\right)}$$

Multiply 16 times 10.

$$x=\frac{-4±\sqrt{16+160}}{2\left(-4\right)}$$

Add 16 to 160.

$$x=\frac{-4±\sqrt{176}}{2\left(-4\right)}$$

Take the square root of 176.

$$x=\frac{-4±4\sqrt{11}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-4±4\sqrt{11}}{-8}$$

Now solve the equation x=\frac{-4±4\sqrt{11}}{-8} when ± is plus. Add -4 to 4\sqrt{11}.

$$x=\frac{4\sqrt{11}-4}{-8}$$

Divide -4+4\sqrt{11} by -8.

$$x=\frac{1-\sqrt{11}}{2}$$

Now solve the equation x=\frac{-4±4\sqrt{11}}{-8} when ± is minus. Subtract 4\sqrt{11} from -4.

$$x=\frac{-4\sqrt{11}-4}{-8}$$

Divide -4-4\sqrt{11} by -8.

$$x=\frac{\sqrt{11}+1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1-\sqrt{11}}{2} for x_{1} and \frac{1+\sqrt{11}}{2} for x_{2}.

$$-4x^{2}+4x+10=-4\left(x-\frac{1-\sqrt{11}}{2}\right)\left(x-\frac{\sqrt{11}+1}{2}\right)$$