Solve 6x^2+3x+2+5x-3x-4

$$6 x ^ { 2 } + 3 x + 2 + 5 x - 3 x - 4$$

Evaluate

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

Solution Steps

Combine 3x and 5x to get 8x.

$$6x^{2}+8x+2-3x-4$$

Combine 8x and -3x to get 5x.

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

Subtract 4 from 2 to get -2.

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

Factor

$$6\left(x-\frac{-\sqrt{73}-5}{12}\right)\left(x-\frac{\sqrt{73}-5}{12}\right)$$

Steps Using the Quadratic Formula

Combine 3x and 5x to get 8x.

$$factor(6x^{2}+8x+2-3x-4)$$

Combine 8x and -3x to get 5x.

$$factor(6x^{2}+5x+2-4)$$

Subtract 4 from 2 to get -2.

$$factor(6x^{2}+5x-2)$$

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.

$$6x^{2}+5x-2=0$$

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.

$$x=\frac{-5±\sqrt{5^{2}-4\times 6\left(-2\right)}}{2\times 6}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 6\left(-2\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-5±\sqrt{25-24\left(-2\right)}}{2\times 6}$$

Multiply -24 times -2.

$$x=\frac{-5±\sqrt{25+48}}{2\times 6}$$

Add 25 to 48.

$$x=\frac{-5±\sqrt{73}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-5±\sqrt{73}}{12}$$

Now solve the equation x=\frac{-5±\sqrt{73}}{12} when ± is plus. Add -5 to \sqrt{73}.

$$x=\frac{\sqrt{73}-5}{12}$$

Now solve the equation x=\frac{-5±\sqrt{73}}{12} when ± is minus. Subtract \sqrt{73} from -5.

$$x=\frac{-\sqrt{73}-5}{12}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5+\sqrt{73}}{12} for x_{1} and \frac{-5-\sqrt{73}}{12} for x_{2}.

$$6x^{2}+5x-2=6\left(x-\frac{\sqrt{73}-5}{12}\right)\left(x-\frac{-\sqrt{73}-5}{12}\right)$$