Solve x+3x^2+4x-1

$$x + 3 x ^ { 2 } + 4 x - 1$$

Factor

$$3\left(x-\frac{-\sqrt{37}-5}{6}\right)\left(x-\frac{\sqrt{37}-5}{6}\right)$$

Steps Using the Quadratic Formula

Combine x and 4x to get 5x.

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

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.

$$3x^{2}+5x-1=0$$

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.

$$x=\frac{-5±\sqrt{5^{2}-4\times 3\left(-1\right)}}{2\times 3}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 3\left(-1\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-5±\sqrt{25-12\left(-1\right)}}{2\times 3}$$

Multiply -12 times -1.

$$x=\frac{-5±\sqrt{25+12}}{2\times 3}$$

Add 25 to 12.

$$x=\frac{-5±\sqrt{37}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-5±\sqrt{37}}{6}$$

Now solve the equation x=\frac{-5±\sqrt{37}}{6} when ± is plus. Add -5 to \sqrt{37}.

$$x=\frac{\sqrt{37}-5}{6}$$

Now solve the equation x=\frac{-5±\sqrt{37}}{6} when ± is minus. Subtract \sqrt{37} from -5.

$$x=\frac{-\sqrt{37}-5}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5+\sqrt{37}}{6} for x_{1} and \frac{-5-\sqrt{37}}{6} for x_{2}.

$$3x^{2}+5x-1=3\left(x-\frac{\sqrt{37}-5}{6}\right)\left(x-\frac{-\sqrt{37}-5}{6}\right)$$

Evaluate

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

Solution Steps

Combine x and 4x to get 5x.

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