Solve x^2+x+12x-5

$$x ^ { 2 } + x + 12 x - 5$$

Factor

$$\left(x-\frac{-3\sqrt{21}-13}{2}\right)\left(x-\frac{3\sqrt{21}-13}{2}\right)$$

Steps Using the Quadratic Formula

Combine x and 12x to get 13x.

$$factor(x^{2}+13x-5)$$

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.

$$x^{2}+13x-5=0$$

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.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-5\right)}}{2}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-5\right)}}{2}$$

Multiply -4 times -5.

$$x=\frac{-13±\sqrt{169+20}}{2}$$

Add 169 to 20.

$$x=\frac{-13±\sqrt{189}}{2}$$

Take the square root of 189.

$$x=\frac{-13±3\sqrt{21}}{2}$$

Now solve the equation x=\frac{-13±3\sqrt{21}}{2} when ± is plus. Add -13 to 3\sqrt{21}.

$$x=\frac{3\sqrt{21}-13}{2}$$

Now solve the equation x=\frac{-13±3\sqrt{21}}{2} when ± is minus. Subtract 3\sqrt{21} from -13.

$$x=\frac{-3\sqrt{21}-13}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-13+3\sqrt{21}}{2} for x_{1} and \frac{-13-3\sqrt{21}}{2} for x_{2}.

$$x^{2}+13x-5=\left(x-\frac{3\sqrt{21}-13}{2}\right)\left(x-\frac{-3\sqrt{21}-13}{2}\right)$$

Evaluate

$$x^{2}+13x-5$$

Solution Steps

Combine x and 12x to get 13x.

$$x^{2}+13x-5$$