Solve 3x^2+6x-3-x^2-5x-1

$$3 x ^ { 2 } + 6 x - 3 - x ^ { 2 } - 5 x - 1$$

Evaluate

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

Solution Steps

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

$$2x^{2}+6x-3-5x-1$$

Combine 6x and -5x to get x.

$$2x^{2}+x-3-1$$

Subtract 1 from -3 to get -4.

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

Factor

$$2\left(x-\frac{-\sqrt{33}-1}{4}\right)\left(x-\frac{\sqrt{33}-1}{4}\right)$$

Steps Using the Quadratic Formula

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

$$factor(2x^{2}+6x-3-5x-1)$$

Combine 6x and -5x to get x.

$$factor(2x^{2}+x-3-1)$$

Subtract 1 from -3 to get -4.

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

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.

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

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.

$$x=\frac{-1±\sqrt{1^{2}-4\times 2\left(-4\right)}}{2\times 2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 2\left(-4\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-1±\sqrt{1-8\left(-4\right)}}{2\times 2}$$

Multiply -8 times -4.

$$x=\frac{-1±\sqrt{1+32}}{2\times 2}$$

Add 1 to 32.

$$x=\frac{-1±\sqrt{33}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-1±\sqrt{33}}{4}$$

Now solve the equation x=\frac{-1±\sqrt{33}}{4} when ± is plus. Add -1 to \sqrt{33}.

$$x=\frac{\sqrt{33}-1}{4}$$

Now solve the equation x=\frac{-1±\sqrt{33}}{4} when ± is minus. Subtract \sqrt{33} from -1.

$$x=\frac{-\sqrt{33}-1}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1+\sqrt{33}}{4} for x_{1} and \frac{-1-\sqrt{33}}{4} for x_{2}.

$$2x^{2}+x-4=2\left(x-\frac{\sqrt{33}-1}{4}\right)\left(x-\frac{-\sqrt{33}-1}{4}\right)$$