Solve 3x^2-2-11x^2+11+4x

$$3 x ^ { 2 } - 2 - 11 x ^ { 2 } + 11 + 4 x$$

Evaluate

$$9+4x-8x^{2}$$

Solution Steps

Combine 3x^{2} and -11x^{2} to get -8x^{2}.

$$-8x^{2}-2+11+4x$$

Add -2 and 11 to get 9.

$$-8x^{2}+9+4x$$

Factor

$$-8\left(x-\frac{1-\sqrt{19}}{4}\right)\left(x-\frac{\sqrt{19}+1}{4}\right)$$

Steps Using the Quadratic Formula

Combine 3x^{2} and -11x^{2} to get -8x^{2}.

$$factor(-8x^{2}-2+11+4x)$$

Add -2 and 11 to get 9.

$$factor(-8x^{2}+9+4x)$$

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.

$$-8x^{2}+4x+9=0$$

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.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-8\right)\times 9}}{2\left(-8\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-8\right)\times 9}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-4±\sqrt{16+32\times 9}}{2\left(-8\right)}$$

Multiply 32 times 9.

$$x=\frac{-4±\sqrt{16+288}}{2\left(-8\right)}$$

Add 16 to 288.

$$x=\frac{-4±\sqrt{304}}{2\left(-8\right)}$$

Take the square root of 304.

$$x=\frac{-4±4\sqrt{19}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-4±4\sqrt{19}}{-16}$$

Now solve the equation x=\frac{-4±4\sqrt{19}}{-16} when ± is plus. Add -4 to 4\sqrt{19}.

$$x=\frac{4\sqrt{19}-4}{-16}$$

Divide -4+4\sqrt{19} by -16.

$$x=\frac{1-\sqrt{19}}{4}$$

Now solve the equation x=\frac{-4±4\sqrt{19}}{-16} when ± is minus. Subtract 4\sqrt{19} from -4.

$$x=\frac{-4\sqrt{19}-4}{-16}$$

Divide -4-4\sqrt{19} by -16.

$$x=\frac{\sqrt{19}+1}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1-\sqrt{19}}{4} for x_{1} and \frac{1+\sqrt{19}}{4} for x_{2}.

$$-8x^{2}+4x+9=-8\left(x-\frac{1-\sqrt{19}}{4}\right)\left(x-\frac{\sqrt{19}+1}{4}\right)$$