Solve 2x-9+3x^2-2-4x

$$2 x - 9 + 3 x ^ { 2 } - 2 - 4 x$$

Evaluate

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

Solution Steps

Subtract 2 from -9 to get -11.

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

Combine 2x and -4x to get -2x.

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

Factor

$$3\left(x-\frac{1-\sqrt{34}}{3}\right)\left(x-\frac{\sqrt{34}+1}{3}\right)$$

Steps Using the Quadratic Formula

Subtract 2 from -9 to get -11.

$$factor(2x-11+3x^{2}-4x)$$

Combine 2x and -4x to get -2x.

$$factor(-2x-11+3x^{2})$$

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.

$$3x^{2}-2x-11=0$$

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.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 3\left(-11\right)}}{2\times 3}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 3\left(-11\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-2\right)±\sqrt{4-12\left(-11\right)}}{2\times 3}$$

Multiply -12 times -11.

$$x=\frac{-\left(-2\right)±\sqrt{4+132}}{2\times 3}$$

Add 4 to 132.

$$x=\frac{-\left(-2\right)±\sqrt{136}}{2\times 3}$$

Take the square root of 136.

$$x=\frac{-\left(-2\right)±2\sqrt{34}}{2\times 3}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{34}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{2±2\sqrt{34}}{6}$$

Now solve the equation x=\frac{2±2\sqrt{34}}{6} when ± is plus. Add 2 to 2\sqrt{34}.

$$x=\frac{2\sqrt{34}+2}{6}$$

Divide 2+2\sqrt{34} by 6.

$$x=\frac{\sqrt{34}+1}{3}$$

Now solve the equation x=\frac{2±2\sqrt{34}}{6} when ± is minus. Subtract 2\sqrt{34} from 2.

$$x=\frac{2-2\sqrt{34}}{6}$$

Divide 2-2\sqrt{34} by 6.

$$x=\frac{1-\sqrt{34}}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{34}}{3} for x_{1} and \frac{1-\sqrt{34}}{3} for x_{2}.

$$3x^{2}-2x-11=3\left(x-\frac{\sqrt{34}+1}{3}\right)\left(x-\frac{1-\sqrt{34}}{3}\right)$$