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

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

Evaluate

$$3\left(x^{2}+x-1\right)$$

Solution Steps

Combine x and 4x to get 5x.

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

Combine 5x and -2x to get 3x.

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

Factor

$$3\left(x-\frac{-\sqrt{5}-1}{2}\right)\left(x-\frac{\sqrt{5}-1}{2}\right)$$

Steps Using the Quadratic Formula

Combine x and 4x to get 5x.

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

Combine 5x and -2x to get 3x.

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

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.

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

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.

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

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 3\left(-3\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-3±\sqrt{9-12\left(-3\right)}}{2\times 3}$$

Multiply -12 times -3.

$$x=\frac{-3±\sqrt{9+36}}{2\times 3}$$

Add 9 to 36.

$$x=\frac{-3±\sqrt{45}}{2\times 3}$$

Take the square root of 45.

$$x=\frac{-3±3\sqrt{5}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-3±3\sqrt{5}}{6}$$

Now solve the equation x=\frac{-3±3\sqrt{5}}{6} when ± is plus. Add -3 to 3\sqrt{5}.

$$x=\frac{3\sqrt{5}-3}{6}$$

Divide -3+3\sqrt{5} by 6.

$$x=\frac{\sqrt{5}-1}{2}$$

Now solve the equation x=\frac{-3±3\sqrt{5}}{6} when ± is minus. Subtract 3\sqrt{5} from -3.

$$x=\frac{-3\sqrt{5}-3}{6}$$

Divide -3-3\sqrt{5} by 6.

$$x=\frac{-\sqrt{5}-1}{2}$$

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

$$3x^{2}+3x-3=3\left(x-\frac{\sqrt{5}-1}{2}\right)\left(x-\frac{-\sqrt{5}-1}{2}\right)$$